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17
Feb

Scientists track nighttime bird migration using weather radar


Discovery
Scientists track nighttime bird migration using weather radar

Weather radar offers new view of nocturnal bird flight

 sandhill cranes flying

Researchers studied sandhill cranes, among the largest migrating birds in North America.
Credit and Larger Version

February 16, 2016

Using recently developed techniques for analyzing Doppler weather radar data, researchers looked at the impediments — crosswinds and oceans — facing nighttime-migrating birds in eastern North America.

The migrants drifted sideways on crosswinds, the scientists found, but compensated for that drift near the Atlantic coast.

Coastal migrating birds’ ability to compensate for wind drift increased through the night, but no strong changes were observed at inland sites. The behavior suggests that birds adapt in flight and compensate for wind drift near coastal areas.

Weather radar tracks bird migration

“The research has taken an innovative approach in showing how existing weather radar systems can be used to investigate the behavior of migrating birds,” said Liz Blood, program director in the National Science Foundation (NSF) Division of Environmental Biology, which funded the research.

Blood said the ability to use the U.S. weather radar network to track migrating birds “opens new opportunities to study — in real-time — billions of birds during their migrations.”

Kyle Horton and Phillip Stepanian of the University of Oklahoma developed an application for observing migrating birds during nighttime flight.

Jeffrey Kelly of the Oklahoma Biological Survey and the University of Oklahoma and Cornell University’s Benjamin Van Doren, Wesley Hochachka and Andrew Farnsworth were also involved in the research.

The results are published in the current issue of the journal Scientific Reports.

“Until now, no studies have documented this large-scale phenomenon using weather radars,” Horton said. “Our analyses are based on detection of millions of migrating birds, as many as five million on a single night.”

The researchers looked at strategies of nocturnally migrating birds using Doppler polarimetric radars at three coastal and three inland sites in the eastern U.S. during the autumns of 2013 and 2014.

Radars collected data every five to 10 minutes, yielding approximately 1.6 million samples from 55 nights.

Birds drift sideways, then compensate

The results show a greater propensity of birds to drift sideways at inland sites; birds flying near the Atlantic coast increasingly oriented and tracked westward away from the coast.

The prediction that migrating birds compensate more for drift when encountering a migration barrier is consistent with the results, the scientists said.

The research indicates that at a regional scale, in a regularly and heavily traveled airspace of the bird migration system, birds routinely fly in crosswind conditions and have managed to compensate for such conditions, according to Kelly.

Migrants likely know their location relative to migration barriers while in flight, he said, and actively assess the degree to which they need to compensate for wind.

“The increasing automation of radar analysis will enable further exploration of U.S. weather radar data to achieve real-time monitoring of billions of birds during their migrations,” Kelly said.

The U.S. weather radar network offers the largest sensor array worldwide for monitoring animal migrations, the scientists said, including birds, bats and insects.

Investigators

Eli Bridge

Le Gruenwald

Jeffrey Kelly

Phillip Chilson

Valliappa Lakshmanan

Related Institutions/Organizations

University of Oklahoma Norman Campus

Related Programs

MacroSystems Biology and Early NEON Science

Related Awards

#1340921 EAGER: Advancing Biological Interpretations of Radar Data

Total Grants

$301,641

14
Oct

Research sheds new light on 150-year-old dinosaur temperature debate


Press Release 15-127
Research sheds new light on 150-year-old dinosaur temperature debate

Evidence shows some dinosaurs elevated their body temperatures using heat sources such as the sun

artist rendition of a dinosaur with eggs and two babies

Scientists used a new technique to find out dinosaurs’ body temperatures based on their eggshells.
Credit and Larger Version

October 13, 2015

Were dinosaurs fast, aggressive hunters like those in the movie “Jurassic World”? Or did they have lower metabolic rates that made them more like today’s alligators and crocodiles?

For 150 years, scientists have debated the nature of dinosaurs’ body temperatures and how they influenced activity levels.

Research by National Science Foundation (NSF)-funded scientists, including John Eiler of the California Institute of Technology, indicates that some dinosaurs had the capacity to elevate their body temperatures using heat sources in the environment, such as the sun.

The researchers also believe the animals were probably more active than modern-day alligators and crocodiles, which can be energetic, but only for brief spurts.

The evidence shows that some dinosaurs had lower body temperatures than modern birds, their only living relatives, and were probably less active.

The research results are published today in the journal Nature Communications.

“These scientists used a relatively new isotope analysis technique on fossil eggshells to investigate thermal regulation in non-avian dinosaurs,” says Rich Lane, program director in NSF’s Division of Earth Sciences, which funded the research. “Comparing the results to modern birds sheds light on the evolution of this trait.”

Led by Robert Eagle, a researcher at UCLA, the scientists examined fossilized dinosaur eggshells from Argentina and Mongolia. Analyzing the shells’ chemistry allowed them to determine the temperatures at which the eggshells formed.

“This technique tells you about the internal body temperature of the female dinosaur when she was ovulating,” says Aradhna Tripati, a co-author of the study and a UCLA geologist. “This presents the first direct measurements of theropod body temperatures.”

The Argentine eggshells, which are approximately 80 million years old, are from large, long-necked titanosaur sauropods, members of a family that includes the largest animals to roam the Earth–relatives of Brontosaurus and Diplodocus.

The shells from Mongolia’s Gobi desert, 71 million to 75 million years old, are from oviraptorid theropods, much smaller dinosaurs that were related to Tyrannosaurus rex and birds.

Sauropods’ body temperatures were warm–approximately 100 degrees Fahrenheit. The smaller dinosaurs had substantially lower temperatures, probably below 90 degrees.

Warm-blooded animals, or endotherms, produce heat internally and maintain their body temperatures, regardless of the temperature of their environment; they do so mainly through metabolism. Humans and other mammals fall into this category.

Cold-blooded animals, or ectotherms, including alligators, crocodiles and lizards, rely on external environmental heat sources to regulate their body temperatures. Lizards, for example, often sit on rocks in the sun to absorb heat, which allows them to be active.

Scientists have long debated whether dinosaurs were endotherms or ectotherms. The research indicates that the answer could lie somewhere in between.

“The temperatures we measured suggest that some dinosaurs were not fully endotherms like modern birds,” Eagle says. “They may have been intermediate–somewhere between modern alligators and crocodiles, and modern birds. That’s the implication for the oviraptorid theropods.

“This could mean that they produced some heat internally and elevated their body temperatures above that of the environment, but didn’t maintain as high temperatures or as controlled temperatures as modern birds. If dinosaurs were endothermic to a degree, they had more capacity to run around searching for food than alligators would.”

The researchers also analyzed fossil soils, including minerals that formed in the upper layers of soils on which the oviraptorid theropod nests were built. That enabled them to estimate that the environmental temperature in Mongolia shortly before the dinosaurs went extinct was approximately 79 degrees Fahrenheit.

“The oviraptorid dinosaur body temperatures were higher than the environmental temperatures–suggesting they were not truly cold-blooded, but intermediate,” Tripati says.

Eagle, Tripati and their colleagues initially looked at modern eggshells from 13 bird species and nine reptiles to establish their ability to determine body temperatures from the chemistry of eggshells.

The researchers measured, in calcium carbonate minerals, the subtle differences in the abundance of chemical bonding between two rare, heavy isotopes: carbon-13 and oxygen-18.

They studied the extent to which these heavy isotopes clustered together using a mass spectrometer–a technique that enabled them to determine mineral formation temperatures. Minerals forming inside colder bodies have more clustering of isotopes.

The scientists analyzed six fossilized eggshells from Argentina, three of which were well-preserved, and 13 eggshells from Mongolia’s Gobi Desert, again selecting three that are well-preserved.

The answers offered new insights into dinosaurs’ body temperatures–and a long-standing debate.

Co-authors of the journal paper are also with the University of Alabama, Tuscaloosa; Germany’s University of Mainz; Columbia University; California State University, Fullerton; California State University, Los Angeles; Orcas Island Historical Museum; CONICET, Argentina; Boise State University; University of Utah; and the Natural History Museum of Los Angeles County.

-NSF-

Media Contacts

Cheryl Dybas, NSF, (703) 292-7734, cdybas@nsf.gov

Stuart Wolpert, UCLA, (310) 206-0511, swolpert@support.ucla.edu

Related Websites
NSF Grant: Insights into Dinosaur Body Temperatures, Physiology, and Environments from Clumped Isotope Thermometry: http://www.nsf.gov/awardsearch/showAward?AWD_ID=1024929HistoricalAwards=false

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2015, its budget is $7.3 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives about 48,000 competitive proposals for funding, and makes about 11,000 new funding awards. NSF also awards about $626 million in professional and service contracts yearly.

 Get News Updates by Email 

Useful NSF Web Sites:

NSF Home Page: http://www.nsf.gov
NSF News: http://www.nsf.gov/news/
For the News Media: http://www.nsf.gov/news/newsroom.jsp
Science and Engineering Statistics: http://www.nsf.gov/statistics/
Awards Searches: http://www.nsf.gov/awardsearch/

 

17
Jun

Big dinosaurs steered clear of the tropics


Press Release 15-067
Big dinosaurs steered clear of the tropics

Climate swings lasting millions of years too much for dinos

Some 212 million years ago, landscapes weren’t all dinosaur-friendly: dry, hot, with wildfires.
Credit and Larger Version

June 15, 2015

For more than 30 million years after dinosaurs first appeared, they remained inexplicably rare near the equator, where only a few small-bodied meat-eating dinosaurs made a living.

The long absence at low latitudes has been one of the great, unanswered questions about the rise of the dinosaurs.

Now the mystery has a solution, according to scientists who pieced together a detailed picture of the climate and ecology more than 200 million years ago at Ghost Ranch in northern New Mexico, a site rich with fossils.

The findings, reported today in the journal Proceedings of the National Academy of Sciences (PNAS), show that the tropical climate swung wildly with extremes of drought and intense heat.

Wildfires swept the landscape during arid regimes and reshaped the vegetation available for plant-eating animals.

“Our data suggest it was not a fun place,” says scientist Randall Irmis of the University of Utah.

“It was a time of climate extremes that went back and forth unpredictably. Large, warm-blooded dinosaurian herbivores weren’t able to exist close to the equator–there was not enough dependable plant food.”

The study, led by geochemist Jessica Whiteside, now of the University of Southampton, is the first to provide a detailed look at climate and ecology during the emergence of the dinosaurs.

Atmospheric carbon dioxide levels then were four to six times current levels. “If we continue along our present course, similar conditions in a high-CO2 world may develop, and suppress low-latitude ecosystems,” Irmis says.

“These scientists have developed a new explanation for the perplexing near-absence of dinosaurs in late Triassic [the Triassic was between 252 million and 201 million years ago] equatorial settings,” says Rich Lane, program director in the National Science Foundation’s (NSF) Division of Earth Sciences, which funded the research.

“That includes rapid vegetation changes related to climate fluctuations between arid and moist climates and the resulting extensive wildfires of the time.”

Reconstructing the deep past

The earliest known dinosaur fossils, found in Argentina, date from around 230 million years ago.

Within 15 million years, species with different diets and body sizes had evolved and were abundant except in tropical latitudes. There the only dinosaurs were small carnivores. The pattern persisted for 30 million years after the first dinosaurs appeared.

The scientists focused on Chinle Formation rocks, which were deposited by rivers and streams between 205 and 215 million years ago at Ghost Ranch (perhaps better known as the place where artist Georgia O’Keeffe lived and painted for much of her career).

The multi-colored rocks of the Chinle Formation are a common sight on the Colorado Plateau at places such as the Painted Desert at Petrified Forest National Park in Arizona.

In ancient times, North America and other land masses were bound together in the supercontinent Pangea. The Ghost Ranch site stood close to the equator, at roughly the same latitude as present-day southern India.

The researchers reconstructed the deep past by analyzing several kinds of data: from fossils, charcoal left by ancient wildfires, stable isotopes from organic matter, and carbonate nodules that formed in ancient soils.

Fossilized bones, pollen grains and fern spores revealed the types of animals and plants living at different times, marked by layers of sediment.

Dinosaurs remained rare among the fossils, accounting for less than 15 percent of vertebrate animal remains.

They were outnumbered in diversity, abundance and body size by reptiles known as pseudosuchian archosaurs, the lineage that gave rise to crocodiles and alligators.

The sparse dinosaurs consisted mostly of small, carnivorous theropods.

Big, long-necked dinosaurs, or sauropodomorphs–already the dominant plant-eaters at higher latitudes–did not exist at the study site nor any other low-latitude site in the Pangaea of that time, as far as the fossil record shows.

Abrupt changes in climate left a record in the abundance of different types of pollen and fern spores between sediment layers.

Fossilized organic matter from decaying plants provided another window on climate shifts. Changes in the ratio of stable isotopes of carbon in the organic matter bookmarked times when plant productivity declined during extended droughts.

Drought and fire

Wildfire temperatures varied drastically, the researchers found, consistent with a fluctuating environment in which the amount of combustible plant matter rose and fell over time.

The researchers estimated the intensity of wildfires using bits of charcoal recovered in sediment layers.

The overall picture is that of a climate punctuated by extreme shifts in precipitation and in which plant die-offs fueled hotter fires. That in turn killed more plants, damaged soils and increased erosion.

Atmospheric carbon dioxide levels, calculated from stable isotope analyses of soil carbonate and preserved organic matter, rose from about 1,200 parts per million (ppm) at the base of the section, to about 2,400 ppm near the top.

At these high CO2 concentrations, climate models predict more frequent and more extreme weather fluctuations consistent with the fossil and charcoal evidence.

Continuing shifts between extremes of dry and wet likely prevented the establishment of the dinosaur-dominated communities found in the fossil record at higher latitudes across South America, Europe, and southern Africa, where aridity and temperatures were less extreme and humidity was consistently higher.

Resource-limited conditions could not support a diverse community of fast-growing, warm-blooded, large dinosaurs, which require a productive and stable environment to thrive.

“The conditions would have been something similar to the arid western United States today, although there would have been trees and smaller plants near streams and rivers, and forests during humid times,” says Whiteside.

“The fluctuating and harsh climate with widespread wildfires meant that only small two-legged carnivorous dinosaurs could survive.”

-NSF-

Media Contacts

Cheryl Dybas, NSF, (703) 292-7734, cdybas@nsf.gov

Joe Rojas-Burke, University of Utah, (801) 585-6861, joe.rojas@utah.edu

Related Websites
NSF Grant: Collaborative Research: An Exceptional Window into Late Triassic Terrestrial Ecosystems from the Western United States: http://www.nsf.gov/awardsearch/showAward?AWD_ID=1349650HistoricalAwards=false

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2015, its budget is $7.3 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives about 48,000 competitive proposals for funding, and makes about 11,000 new funding awards. NSF also awards about $626 million in professional and service contracts yearly.

 Get News Updates by Email 

Useful NSF Web Sites:

NSF Home Page: http://www.nsf.gov
NSF News: http://www.nsf.gov/news/
For the News Media: http://www.nsf.gov/news/newsroom.jsp
Science and Engineering Statistics: http://www.nsf.gov/statistics/
Awards Searches: http://www.nsf.gov/awardsearch/

 

17
Jan

Tiny plant fossils offer window into Earth’s landscape millions of years ago


Press Release 15-003
Tiny plant fossils offer window into Earth’s landscape millions of years ago

Fossilized plant pieces tell a detailed story of our planet 50 million years ago

“Hemispherical” photograph of an open habitat at Rincon de la Vieja National Park, Costa Rica.
Credit and Larger Version

January 15, 2015

Minuscule, fossilized pieces of plants tell a detailed story of what Earth looked like 50 million years ago.

Researchers have discovered a way of determining density of trees, shrubs and bushes in locations over time–based on clues in the cells of plant fossils preserved in rocks and soil.

Tree density directly affects precipitation, erosion, animal behavior and a host of other factors in the natural world. Quantifying vegetation structure throughout time could shed light on how Earth’s ecosystems have changed over millions of years.

“Knowing an area’s vegetation structure and the arrangement of leaves on the Earth’s surface is key to understanding the terrestrial ecosystem,” says Regan Dunn, a paleontologist at the University of Washington’s Burke Museum of Natural History and Culture. “It’s the context in which all land-based organisms live, but we didn’t have a way to measure it until now.”

The findings are published in this week’s issue of the journal Science.

New method offers window into distant past

“The new methodology provides a high-resolution lens for viewing the structure of ecosystems over the deep history of our planet,” says Alan Tessier, acting director of the National Science Foundation’s (NSF) Division of Environmental Biology, which funded the research along with NSF’s Division of Earth Sciences.

“This capability will advance the field of paleoecology and greatly improve our understanding of how future climate change will reshape ecosystems.”

The team focused its fieldwork on several sites in Patagonia, which have some of the best preserved fossils in the world.

For years, paleontologists have painstakingly collected fossils from these sites and worked to precisely determine their ages using radiometric dating. The new study builds on this growing body of knowledge.

In Patagonia and other places, scientists have some idea based on records of fossilized pollen and leaves what species of plants were alive at given periods in history.

For example, the team’s previous work documented vegetation composition for this area.

But there hasn’t been a way to precisely quantify vegetation openness, aside from general speculations of open or bare habitats, as opposed to closed or tree-covered habitats.

“These researchers have developed a new method for reconstructing paleo-vegetation structure in open versus dense forests using plant biosilica, likely to be widely found in the fossil record,” says Chris Liu, program director in NSF’s Division of Earth Sciences.

“Now we have a tool to look at a lot of important intervals in our history where we don’t know what happened to the structure of vegetation,” adds Dunn, such as the period just after the mass extinction that killed the dinosaurs.

“Vegetation structure links all aspects of modern ecosystems, from soil moisture to primary productivity to global climate,” says paper co-author Caroline Stromberg, a curator of paleobotany at the Burke Museum.

“Using this method, we can finally quantify in detail how Earth’s plant and animal communities have responded to climate change over millions of years, vital for forecasting how ecosystems will change under predicted future climate scenarios.”

Plant cell patterns change with sun exposure

Work by other scientists has shown that the cells found in a plant’s outermost layer, called the epidermis, change in size and shape depending on how much sun it’s exposed to while its leaves develop.

For example, the cells of a leaf that grow in deeper shade will be larger and curvier than the cells of leaves that develop in less covered areas.

Dunn and collaborators found that these cell patterns, indicating growth in shade or sun, similarly show up in some plant fossils.

When a plant’s leaves fall to the ground and decompose, tiny silica particles inside the plants, called phytoliths, remain as part of the soil layer.

The phytoliths were found to represent epidermal cell shapes and sizes, indicating whether the plant grew in a shady or open area.

The researchers decided to check their hypothesis by testing it in a modern setting: Costa Rica.

Dunn took soil samples from sites in Costa Rica that varied from covered rainforests to open savannas to woody shrublands.

She also took photos looking directly up at the tree canopy (or lack thereof) at each site, noting the total vegetation coverage.

Back in the lab, she extracted the phytoliths from each soil sample and measured them under the microscope.

When compared with tree coverage estimated from the corresponding photos, Dunn and co-authors found that the curves and sizes of the cells directly related to how shady their environment was.

“Leaf area index” and plant cell structures compared

The researchers characterized the amount of shade as “leaf area index,” a standard way of measuring vegetation over a specific area.

Testing this relationship between leaf area index and plant cell structures in modern environments allowed the scientists to develop an equation that can be used to predict vegetation openness at any time in the past, provided there are preserved plant fossils.

“Leaf area index is a well-known variable for ecologists, climate scientists and modelers, but no one’s ever been able to imagine how you could reconstruct tree coverage in the past–and now we can,” says co-author Richard Madden of the University of Chicago.

“We should be able to reconstruct leaf area index by using all kinds of fossil plant preservation, not just phytoliths. Once that is demonstrated, then the places in the world where we can reconstruct this will increase.”

When Dunn and co-authors applied their method to 40-million-year-old phytoliths from Patagonia, they found something surprising–vegetation was extremely open, similar to a shrubland today. The appearance of these very open habitats coincided with major changes in fauna.

The paleobiologists plan to test the relationship between vegetation coverage and plant cell structure in other regions around the world.

They also hope to find other types of plant fossils that hold the same information at the cellular level as do phytoliths.

Paper co-authors are Matthew Kohn of Boise State University and Alfredo Carlini of Universidad Nacional de La Plata in Argentina.

In addition to NSF, the research was funded by the Geological Society of America, the University of Washington Biology Department and the Burke Museum.

-NSF-

Media Contacts

Cheryl Dybas, NSF, (703) 292-7734, cdybas@nsf.gov

Michelle Ma, University of Washington, (206) 543-2580, mcma@uw.edu

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2014, its budget is $7.2 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives about 50,000 competitive requests for funding, and makes about 11,500 new funding awards. NSF also awards about $593 million in professional and service contracts yearly.

 Get News Updates by Email 

Useful NSF Web Sites:

NSF Home Page: http://www.nsf.gov
NSF News: http://www.nsf.gov/news/
For the News Media: http://www.nsf.gov/news/newsroom.jsp
Science and Engineering Statistics: http://www.nsf.gov/statistics/
Awards Searches: http://www.nsf.gov/awardsearch/

 

12
Jan

A free boundary problem arising in the ecological models with N -species

Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M228','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M228View MathML/a be given and let (2.5) hold for some a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M229','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M229View MathML/a. The aim of this section is to prove Theorem 2.1.

3.1 An approximation problem

We will construct an approximation problem of (1.1a)-(1.3). We first construct some approximation functions.

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(3.1)

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a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M243','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M243View MathML/a

Define

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M244','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M244View MathML/a

(3.2)

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(3.3)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M246','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M246View MathML/a

Then according to hypothesis (H)(ii) and (iii), it follows that the vector function
a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M247','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M247View MathML/a is mixed quasimonotone in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M248','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M248View MathML/a with index vector a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M249','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M249View MathML/a, and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M250','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M250View MathML/a

(3.4)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M251','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M251View MathML/a

(3.5)

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(3.6)

where a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M253','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M253View MathML/a. The definition of function a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M254','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M254View MathML/a implies that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M255','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M255View MathML/a

(3.7)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M256','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M256View MathML/a

(3.8)

In addition, it is obvious from (2.5) that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M257','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M257View MathML/a

(3.9)

where a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M258','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M258View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M259','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M259View MathML/a are the closure of a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M260','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M260View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M261','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M261View MathML/a, respectively. Thus by (3.9) and the definition of functions a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M262','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M262View MathML/a, an argument similar to the one used in 16], Lemma 3.2] shows that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M263','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M263View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M264','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M264View MathML/a

(3.10)

We next construct the approximation functions of a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M265','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M265View MathML/a. Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M266','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M266View MathML/a be a sufficiently smooth nonnegative function such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M267','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M267View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M268','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M268View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M269','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M269View MathML/a, and let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M270','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M270View MathML/a be a sufficiently smooth nonnegative function taking values in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M271','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M271View MathML/a such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M272','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M272View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M273','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M273View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M274','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M274View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M275','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M275View MathML/a or a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M276','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M276View MathML/a, and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M277','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M277View MathML/a for all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M278','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M278View MathML/a. Define

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M279','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M279View MathML/a

Then hypothesis (H)(i) and 17], Chapter II] imply that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M280','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M280View MathML/a

(3.11)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M281','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M281View MathML/a

and (2.1) and (2.3) imply that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M282','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M282View MathML/a

(3.12)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M283','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M283View MathML/a

(3.13)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M284','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M284View MathML/a

(3.14)

In addition, for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M285','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M285View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M286','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M286View MathML/a

(3.15)

Employing the above approximation functions, we consider the following approximation
problem:

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M287','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M287View MathML/a

(3.16)

Lemma 3.1

Problem (3.16) has a unique classical solutiona onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M288','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M288View MathML/aina onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M289','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M289View MathML/a, and the following estimates hold:

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M290','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M290View MathML/a

(3.17)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M291','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M291View MathML/a

(3.18)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M292','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M292View MathML/a

(3.19)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M293','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M293View MathML/a

(3.20)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M294','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M294View MathML/a

(3.21)

where constantsa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M295','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M295View MathML/aandCare independent ofε.

Proof

In 15], by using the method of upper and lower solutions, together with the associated monotone
iterations and various estimates, we investigated the existence and uniqueness of
the global piecewise classical solutions of the quasilinear parabolic system with
discontinuous coefficients and continuous delays under various conditions including
mixed quasimonotone property of reaction functions. The same problem was also discussed
for the system with continuous coefficients without time-delay.

It is obvious that problem (3.16) is the special case of 15], problem (1.1)] without discontinuous coefficients and time delays. Hypothesis (H)(ii)
shows that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M296','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M296View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M297','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M297View MathML/a are the coupled weak upper and lower solutions of (3.16) in the sense of 15], Definition 2.2]. By (3.4)-(3.6) and (3.11)-(3.14), we conclude from 15], Theorem 4.1] that problem (3.16) has a unique classical solution a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M298','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M298View MathML/a in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M299','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M299View MathML/a. Furthermore, using (3.1), (3.5), (3.6) and (3.12)-(3.14), the proof similar to that of 16], Lemma 3.3] shows that estimates (3.17) and (3.18) hold.

To prove (3.19), we first fix a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M300','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M300View MathML/a. In view of (3.7), (3.8), (3.10) and (3.15), we find that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M301','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M301View MathML/a is the solution of the following problems for single equation:

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M302','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M302View MathML/a

(3.22)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M303','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M303View MathML/a

(3.23)

By (3.22), (3.23), (3.12) and (3.14), the proof similar to that of 17], Chapter VI, Lemma 3.1] gives (3.19) for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M304','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M304View MathML/a. The similar argument shows that (3.19) holds for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M305','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M305View MathML/a.

We next prove (3.20). For any fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M306','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M306View MathML/a, let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M307','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M307View MathML/a

and let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M308','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M308View MathML/a

By a direct computation we have

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M309','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M309View MathML/a

(3.24)

Then (3.16), (3.2), (3.3) and (3.10) imply that the function a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M310','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M310View MathML/a satisfies

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M311','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M311View MathML/a

(3.25)

where a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M312','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M312View MathML/a.

A double integration by parts gives

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M313','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M313View MathML/a

(3.26)

Thus multiplying the equation in (3.25) by a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M314','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M314View MathML/a, integrating it on a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M315','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M315View MathML/a and using (3.5), (3.6), (3.24) and (3.26), we find that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M316','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M316View MathML/a

Furthermore, by (3.12)-(3.14), (3.19) and Cauchy’s inequality, we deduce that for any a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M317','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M317View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M318','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M318View MathML/a

Choosing a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M319','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M319View MathML/a, we get

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M320','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M320View MathML/a

(3.27)

Consequently,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M321','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M321View MathML/a

This, together with Gronwall’s inequality (see 17], Chapter II, Lemma 5.5]), implies that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M322','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M322View MathML/a

Hence we deduce from (3.25), (3.27), (3.5) and (3.6) that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M323','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M323View MathML/a

which, together with (3.24), yields (3.20) for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M324','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M324View MathML/a.

For any fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M325','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M325View MathML/a, we consider the equality

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M326','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M326View MathML/a

A similar argument gives (3.20) for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M327','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M327View MathML/a. Therefore, (3.20) holds for all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M328','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M328View MathML/a.

It remains to prove (3.21). For each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M329','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M329View MathML/a, since a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M330','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M330View MathML/a is in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M331','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M331View MathML/a, then (3.20) and 17], Chapter 2, formula (3.8)] show that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M332','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M332View MathML/a

Thus (3.21) holds. □

Lemma 3.2

Leta onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M333','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M333View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M334','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M334View MathML/abe the solutions ina onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M335','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M335View MathML/afor problem (3.16) corresponding toa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M336','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M336View MathML/aanda onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M337','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M337View MathML/a, respectively. Then

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M338','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M338View MathML/a

(3.28)

wherea onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M339','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M339View MathML/a.

Proof

Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M340','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M340View MathML/a, and let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M341','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M341View MathML/a be fixed. We see from (3.16) that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M342','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M342View MathML/a

(3.29)

In view of (3.15), we find a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M343','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M343View MathML/a. Multiplying the equation in (3.29) by a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M344','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M344View MathML/a and integrating by parts on a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M345','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M345View MathML/a, we deduce that, for any a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M346','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M346View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M347','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M347View MathML/a

(3.30)

Let us estimate a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M348','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M348View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M349','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M349View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M350','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M350View MathML/a. Since (3.2), (3.3) and (3.10) imply that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M351','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M351View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M352','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M352View MathML/a

then it follows from Cauchy’s inequality that, for any a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M353','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M353View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M354','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M354View MathML/a

(3.31)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M355','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M355View MathML/a

(3.32)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M356','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M356View MathML/a

(3.33)

where

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M357','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M357View MathML/a

According to the definition of function a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M358','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M358View MathML/a, we see that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M359','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M359View MathML/a if a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M360','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M360View MathML/a or a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M361','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M361View MathML/a. Thus by (3.21) we have

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M362','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M362View MathML/a

(3.34)

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M363','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M363View MathML/a

(3.35)

Summing equality (3.30) with respect to l from a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M364','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M364View MathML/a to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M365','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M365View MathML/a, using (3.31)-(3.35) and Minkowski’s inequality, and choosing a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M366','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M366View MathML/a, we then conclude that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M367','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M367View MathML/a

This, together with Gronwall’s inequality, yields

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M368','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M368View MathML/a

Combining the two inequalities above and (3.21) leads us to estimate (3.28). □

3.2 The solutions of the diffraction problem

Proof of Theorem 2.1

We divide the proof into three steps.

Step 1. We prove the global existence of the solutions. Let us discuss the behavior
of the solution a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M369','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M369View MathML/a associated with a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M370','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M370View MathML/a by Theorem 2.1 as a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M371','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M371View MathML/a.

We first see from (3.16) that for any a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M372','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M372View MathML/a and any vector function a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M373','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M373View MathML/a,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M374','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M374View MathML/a

(3.36)

Furthermore, according to estimates (3.17), (3.20), (3.28) and the Arzela-Ascoli theorem, we conclude that there exists a subsequence (we retain
the same notation for it) a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M375','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M375View MathML/a such that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M376','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M376View MathML/a

Thus u is in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M377','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M377View MathML/a, u satisfies the parabolic condition (1.2), and estimates (2.6) and (2.7) hold.

We next show that for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M378','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M378View MathML/a, the sequences a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M379','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M379View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M380','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M380View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M381','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M381View MathML/a converge in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M382','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M382View MathML/a to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M383','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M383View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M384','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M384View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M385','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M385View MathML/a, respectively. Since

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M386','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M386View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M387','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M387View MathML/a

then it follows from (2.2), (3.3) and Lebesgue dominated convergence theorem that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M388','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M388View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M389','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M389View MathML/a

and from (3.10) that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M390','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M390View MathML/a

The similar argument shows that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M391','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M391View MathML/a for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M392','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M392View MathML/a.

Based on the above arguments for sequences a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M393','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M393View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M394','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M394View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M395','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M395View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M396','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M396View MathML/a, by letting a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M397','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M397View MathML/a, we conclude from (3.36) that (2.4) holds.

We also see from (3.20) that there exists a subsequence a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M398','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M398View MathML/a (denoted by a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M399','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M399View MathML/a still) such that for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M400','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M400View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M401','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M401View MathML/a converge weakly in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M402','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M402View MathML/a to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M403','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M403View MathML/a. Recalling that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M404','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M404View MathML/a in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M405','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M405View MathML/a, we deduce a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M406','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M406View MathML/a. These, together with (3.20), imply that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M407','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M407View MathML/a for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M408','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M408View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M409','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M409View MathML/a for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M410','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M410View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M411','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M411View MathML/a for each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M412','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M412View MathML/a, and (2.8) holds. Thus (2.4) implies that u satisfies the equations in (1.1a) and (1.1b) for almost all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M413','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M413View MathML/a and the inner boundary condition (1.3) for almost all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M414','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M414View MathML/a (see 17], Chapter 3, Section 13]). As we have done in the derivation of (3.21), estimate (2.8) yields (2.9).

For fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M415','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M415View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M416','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M416View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M417','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M417View MathML/a satisfies the linear equation

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M418','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M418View MathML/a

where

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M419','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M419View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M420','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M420View MathML/a

Then for any subdomains a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M421','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M421View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M422','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M422View MathML/a satisfying a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M423','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M423View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M424','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M424View MathML/a, we have a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M425','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M425View MathML/a. The parabolic regularity theory shows that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M426','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M426View MathML/a. Hence u satisfies pointwise the equations in (1.1a) and (1.1b) for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M427','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M427View MathML/a. Consequently, u is a solution in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M428','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M428View MathML/a of problem (1.1a)-(1.3) and estimates (2.6)-(2.9) hold.

Step 2. In what follows, we will show that the solution in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M429','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M429View MathML/a for problem (1.1a)-(1.3) is unique and estimates (2.10) and (2.11) hold.

Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M430','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M430View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M431','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M431View MathML/a be the solutions in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M432','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M432View MathML/a corresponding to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M433','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M433View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M434','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M434View MathML/a, respectively. Set a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M435','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M435View MathML/a. Then a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M436','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M436View MathML/a. We choose a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M437','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M437View MathML/a in (2.4) to find

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M438','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M438View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M439','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M439View MathML/a

For each a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M440','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M440View MathML/a, by a subtraction of the above equations for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M441','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M441View MathML/a, we conclude that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M442','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M442View MathML/a

and

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M443','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M443View MathML/a

Then

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M444','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M444View MathML/a

Setting a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M445','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M445View MathML/a and summing the above inequalities with respect to l from a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M446','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M446View MathML/a to a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M447','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M447View MathML/a, we have

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M448','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M448View MathML/a

Again by Gronwall’s inequality we deduce (2.10), which, together with 17], Chapter 2, formula (3.8)], gives (2.11). Therefore the solution in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M449','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M449View MathML/a for problem (1.1a)-(1.3) associated with a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M450','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M450View MathML/a is unique.

Step 3. For a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M451','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M451View MathML/a, we discuss the regularity of u.

For any fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M452','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M452View MathML/a, let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M453','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M453View MathML/a

Then v satisfies

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M454','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M454View MathML/a

We will use the result of 18] to obtain the regularity of v. To do this, we need the estimate of a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M455','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M455View MathML/a for any fixed a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M456','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M456View MathML/a. Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M457','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M457View MathML/a be a smooth function with values between 0 and 1 such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M458','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M458View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M459','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M459View MathML/a or a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M460','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M460View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M461','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M461View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M462','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M462View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M463','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M463View MathML/a, and let

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M464','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M464View MathML/a

For any small enough Δt, consider the equality a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M465','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M465View MathML/a. By employing the formula of integration by parts and 17], Chapter II, formula (4.7)], we get

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M466','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M466View MathML/a

where a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M467','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M467View MathML/a. Some tedious computation and Cauchy’s inequality yield

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M468','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M468View MathML/a

We choose a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M469','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M469View MathML/a and employ (2.8) to find

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M470','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M470View MathML/a

As we have done in the derivation of (3.20), by Gronwall’s inequality we get

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M471','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M471View MathML/a

Consequently,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M472','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M472View MathML/a

By 17], Chapter II, Lemma 4.11], this inequality implies that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M473','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M473View MathML/a

(3.37)

Hence a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M474','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M474View MathML/a. Using (3.37), hypothesis (H)(iii) and 18], Theorem 1.1], we deduce that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M475','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M475View MathML/a is continuous with respect to y in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M476','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M476View MathML/a and in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M477','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M477View MathML/a for almost all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M478','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M478View MathML/a, and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M479','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M479View MathML/a is continuous in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M480','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M480View MathML/a. Since a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M481','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M481View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M482','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M482View MathML/a, then for almost all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M483','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M483View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M484','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M484View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M485','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M485View MathML/a are continuous with respect to x in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M486','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M486View MathML/a and in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M487','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M487View MathML/a. □

The following corollary follows directly from Theorem 2.1.

Corollary 3.3

Assume thata onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M488','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M488View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M489','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M489View MathML/a , satisfy (2.5) and the sequencea onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M490','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M490View MathML/aconverges ina onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M491','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M491View MathML/atoa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M492','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M492View MathML/a. Ifa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M493','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M493View MathML/a, uare the solutions ina onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M494','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M494View MathML/aof (1.1a)-(1.3) corresponding toa onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M495','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M495View MathML/aandφ , respectively, then there exists a subsequence (we retain the same notation for it) a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M496','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M496View MathML/asuch that

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M497','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M497View MathML/a

1
Jan

Measuring the Risks and Causes of Premature Death: Summary of a Workshop

The final version of this book has not been published yet. You can pre-order a copy of the book and we will send it to you when it becomes available. We will not charge you for the book until it ships. Pricing for a pre-ordered book is estimated and subject to change. All backorders will be released at the final established price. As a courtesy, if the price increases by more than $3.00 we will notify you.

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13
Dec

‘Big bang’ of bird evolution mapped by international research team


Press Release 14-170
‘Big bang’ of bird evolution mapped by international research team

Genes reveal histories of bird origins, feathers, flight and song

Peregrine falcon

Peregrine falcons are more closely related to parrots and songbirds than to hawks, eagles, or owls.
Credit and Larger Version

December 11, 2014

The genomes of modern birds tell a story: Today’s winged rulers of the skies emerged and evolved after the mass extinction that wiped out dinosaurs and almost everything else 66 million years ago.

That story is now coming to light, thanks to an international collaboration that has been underway for four years.

The first findings of the Avian Phylogenomics Consortium are being reported nearly simultaneously in 23 papers–eight papers in a special issue this week of Science, and 15 more in Genome Biology, GigaScience and other journals.

The results are funded in part by the National Science Foundation (NSF).

Scientists already knew that the birds that survived the mass extinction experienced a rapid burst of evolution.

But the family tree of modern birds has confused biologists for centuries, and the molecular details of how birds arrived at the spectacular biodiversity of more than 10,000 species was barely known.

How did birds become so diverse?

To resolve these fundamental questions, a consortium led by Guojie Zhang of the National Genebank at BGI in China and the University of Copenhagen; neuroscientist Erich Jarvis of Duke University and the Howard Hughes Medical Institute; and M. Thomas P. Gilbert of the Natural History Museum of Denmark has sequenced, assembled and compared the full genomes of 48 bird species.

The species include the crow, duck, falcon, parakeet, crane, ibis, woodpecker, eagle and others, representing all major branches of modern birds.

“BGI’s strong support and four years of hard work by the entire community have enabled us to answer numerous fundamental questions on an unprecedented scale,” said Zhang.

“This is the largest whole genomic study across a single vertebrate class to date. The success of this project can only be achieved with the excellent collaboration of all the consortium members.”

Added Gilbert, “Although an increasing number of vertebrate genomes are being released, to date no single study has deliberately targeted the full diversity of any major vertebrate group.

“This is what our consortium set out to do. Only with this scale of sampling can scientists truly begin to fully explore the genomic diversity within a full vertebrate class.”

“This is an exciting moment,” said Jarvis. “Lots of fundamental questions now can be resolved with more genomic data from a broader sampling. I got into this project because of my interest in birds as a model for vocal learning and speech production in humans, and it has opened up some amazing new vistas on brain evolution.”

This first round of analyses suggests some remarkable new ideas about bird evolution.

The first flagship paper published in Science presents a well-resolved new family tree for birds, based on whole-genome data.

The second flagship paper describes the big picture of genome evolution in birds.

Six other papers in the special issue of Science report how vocal learning may have independently evolved in a few bird groups and in the human brain’s speech regions; how the sex chromosomes of birds came to be; how birds lost their teeth; how crocodile genomes evolved; and ways in which singing behavior regulates genes in the brain.

New ideas on bird evolution

“This project represents the biggest step forward yet in our understanding of how bird diversity is organized and in time and space,” said paper co-author Scott Edwards, on leave from Harvard University and currently Director of NSF’s Division of Biological Infrastructure.

“Because this information is so fundamental to our understanding of biodiversity, it will help everyone–from birdwatchers to artists to museum curators–better organize knowledge of bird diversity.”

The new bird tree will change the way we think about bird diversity, said Edwards. “The fact that many birds associated with water–loons, herons, penguins, petrels and pelicans–are closely related suggests that adaptations to lakes or seas arose less frequently than we thought.”

Added paper co-author David Mindell, an evolutionary biologist and program director in NSF’s Division of Environmental Biology, “We found strong support for close relationships that might be surprising to many observers.

“Grebes are closely related to flamingos, but not closely related to ducks; falcons are closely related to songbirds and parrots but not closely related to hawks; and swifts are closely related to hummingbirds and not closely related to swallows.”

Genome-scale datasets allowed scientists to “track the sequence of divergence events and their timing with greater precision than previously possible,” said Mindell.

“Most major types of extant birds arose during a 5-10 million year interval at the end of the Cretaceous period and the extinction of non-avian dinosaurs about 66 million years ago.”

It takes a consortium…of 200 scientists, 80 institutions, 20 countries

The Avian Phylogenomics Consortium has so far involved more than 200 scientists from 80 institutions in 20 countries, including the BGI in China, the University of Copenhagen, Duke University, the University of Texas at Austin, the Smithsonian Institution, the Chinese Academy of Sciences, Louisiana State University and others.

Previous attempts to reconstruct the avian family tree using partial DNA sequencing or anatomical and behavioral traits have met with contradiction and confusion.

Because modern birds split into species early and in such quick succession, they did not evolve enough distinct genetic differences at the genomic level to clearly determine their early branching order, the researchers said.

To resolve the timing and relationships of modern birds, consortium scientists used whole-genome DNA sequences to infer the bird species tree.

“In the past, people have been using 10 to 20 genes to try to infer the species relationships,” Jarvis said.

“What we’ve learned from doing this whole-genome approach is that we can infer a somewhat different phylogeny [family tree] than what has been proposed in the past.

“We’ve figured out that protein-coding genes tell the wrong story for inferring the species tree. You need non-coding sequences, including the intergenic regions. The protein-coding sequences, however, tell an interesting story of proteome-wide convergence among species with similar life histories.”

Where did all the birds come from?

This new tree resolves the early branches of Neoaves (new birds) and supports conclusions about relationships that have been long-debated.

For example, the findings support three independent origins of waterbirds.

They also indicate that the common ancestor of core landbirds, which include songbirds, parrots, woodpeckers, owls, eagles and falcons, was an apex predator, which also gave rise to the giant terror birds that once roamed the Americas.

The whole-genome analysis dates the evolutionary expansion of Neoaves to the time of the mass extinction event 66 million years ago.

This contradicts the idea that Neoaves blossomed 10 to 80 million years earlier, as some recent studies have suggested.

Based on this new genomic data, only a few bird lineages survived the mass extinction.

They gave rise to the more than 10,000 Neoaves species that comprise 95 percent of all bird species living with us today.

The freed-up ecological niches caused by the extinction event likely allowed rapid species radiation of birds in less than 15 million years, which explains much of modern bird biodiversity.

For answers, new computational tools needed

Increasingly sophisticated and more affordable genomic sequencing technologies, and the advent of computational tools for reconstructing and comparing whole genomes, have allowed the consortium to resolve these controversies with better clarity than ever before, the researchers said.

With about 14,000 genes per species, the size of the datasets and the complexity of analyzing them required new approaches to computing evolutionary family trees.

These were developed by computer scientists Tandy Warnow at the University of Illinois at Urbana-Champaign, funded by NSF, Siavash Mirarab of the University of Texas at Austin, and Alexis Stamatakis at the Heidelburg Institute for Theoretical Studies.

Their algorithms required the use of parallel processing supercomputers at the Munich Supercomputing Center, the Texas Advanced Computing Center, and the San Diego Supercomputing Center.

“The computational challenges in estimating the avian species tree used around 300 years of CPU time, and some analyses required supercomputers with a terabyte of memory,” Warnow said.

The bird project also had support from the Genome 10K Consortium of Scientists (G10K), an international science community working toward rapidly assessing genome sequences for 10,000 vertebrate species.

“The Avian Genomics Consortium has accomplished the most ambitious and successful project that the G10K Project has joined or endorsed,” said G10K co-leader Stephen O’Brien, who co-authored a commentary on the bird sequencing project in GigaScience.

-NSF-

Media Contacts

Cheryl Dybas, NSF, (703) 292-7734, cdybas@nsf.gov

Karl Bates, Duke University, (919) 681-8054, karl.bates@duke.edu

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2014, its budget is $7.2 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives about 50,000 competitive requests for funding, and makes about 11,500 new funding awards. NSF also awards about $593 million in professional and service contracts yearly.

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9
Sep

Paleontologists discover new species of titanosaurian dinosaur in Tanzania


Press Release 14-115
Paleontologists discover new species of titanosaurian dinosaur in Tanzania

Rare find of sauropod dinosaur skeleton from Africa

Where did Rukwatitan live? This artist’s rendition shows the dinosaur’s likely paleoenvironment.
Credit and Larger Version

September 8, 2014

For video b-roll associated with this discovery, please contact Dena Headlee at dheadlee@nsf.gov.

Paleontologists have identified a new species of titanosaurian, a member of the large-bodied sauropods that thrived during the final period of the dinosaur age, in Tanzania.

Although many fossils of titanosaurians have been discovered around the globe, especially in South America, few have been recovered from the continent of Africa.

The new species, named Rukwatitan bisepultus, was first spotted embedded in a cliff wall in the Rukwa Rift Basin of southwestern Tanzania.

With the help of professional excavators and coal miners, the scientists unearthed vertebrae, ribs, limbs and pelvic bones over the course of several months.

CT scans of the fossils, combined with detailed comparisons with other sauropods, revealed unique features that suggested an animal that was different from previous finds–including those from elsewhere in Africa, according to a paper published today in the Journal of Vertebrate Paleontology.

“This titanosaur finding is rare for Africa, and will help resolve questions about the distribution and regional characteristics of what would later become one of the largest land animals known,” says Paul Filmer, a program director in the National Science Foundation’s (NSF) Division of Earth Sciences, which funded the research.

“Titanosaurians make up the vast majority of known Cretaceous sauropods, and have been found on every continent, yet Africa has so far yielded only four formally recognized members.”

Rukwatitan bisepultus lived approximately 100 million years ago during the middle of the Cretaceous Period.

Titanosaurian sauropods, the group that includes Rukwatitan, were herbivorous dinosaurs known for their iconic large body sizes, long necks and wide stance.

Although not among the largest of titanosaurians, Rukwatitan is estimated to have forelimbs reaching 2 meters and may have weighed as much as several elephants.

“Using traditional and new computational approaches, we were able to place the new species within the family tree of sauropod dinosaurs and determine its uniqueness as a species–and to delineate other species with which it is most closely related,” says lead paper author Eric Gorscak, a biologist at Ohio University.

The dinosaur’s bones exhibit similarities with another titanosaurian, Malawisaurus dixeyi, previously recovered in Malawi.

But the two dinosaurs are distinctly different from one another, and from titanosaurians known from northern Africa, says co-author Patrick O’Connor, an anatomist at Ohio University’s Heritage College of Osteopathic Medicine.

The fossils of middle Cretaceous crocodile relatives from the Rukwa Rift Basin also exhibit distinctive features when compared to forms from elsewhere on the continent.

“There may have been certain environmental features, such as deserts, large waterways and/or mountain ranges, that would have limited the movement of animals and promoted the evolution of regionally distinct faunas,” O’Connor says.

“Only additional data on faunas and paleoenvironments from around the continent will let us further test such hypotheses.”

In addition to providing new data about species evolution in sub-Saharan Africa, the results contribute to fleshing out the portrait of titanosaurians, which lived in habitats across the globe through the end of the Cretaceous period.

Their rise in diversity came in the wake of the decline of another group of sauropods, the diplodocoids, which include the dinosaur Apatosaurus.

“Much of what we know about titanosaurian evolutionary history stems from numerous discoveries in South America–a continent that underwent a steady separation from Africa during the first half of the Cretaceous Period,” Gorscak says.

“With the discovery of Rukwatitan and study of the material in nearby Malawi, we are beginning to fill a significant gap from a large part of the world.”

Co-authors of the paper are Nancy Stevens of the Ohio University Heritage College of Osteopathic Medicine and Eric Roberts of James Cook University of Australia.

The study was also funded by the National Geographic Society, the Ohio University Heritage College of Osteopathic Medicine and the Ohio University Office of the Vice President for Research and Creative Activity.

-NSF-

Media Contacts

Cheryl Dybas, NSF, (703) 292-7734, cdybas@nsf.gov

Andrea Gibson, Ohio University, (740) 597-2166, gibsona@ohio.edu

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2014, its budget is $7.2 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives about 50,000 competitive requests for funding, and makes about 11,500 new funding awards. NSF also awards about $593 million in professional and service contracts yearly.

 Get News Updates by Email 

Useful NSF Web Sites:

NSF Home Page: http://www.nsf.gov
NSF News: http://www.nsf.gov/news/
For the News Media: http://www.nsf.gov/news/newsroom.jsp
Science and Engineering Statistics: http://www.nsf.gov/statistics/
Awards Searches: http://www.nsf.gov/awardsearch/

 

5
Sep

T. Rex times seven: New dinosaur species is discovered in Argentina


Press Release 14-111
T. Rex times seven: New dinosaur species is discovered in Argentina

Drexel researchers uncover immense, remarkably complete dinosaur skeleton; research team includes three NSF Graduate Research Fellows

Kenneth Lacovara surrounded by the skeleton of Dreadnoughtus schrani.
Credit and Larger Version

September 4, 2014

For video b-roll associated with this discovery, please contact Dena Headlee at dheadlee@nsf.gov.

Scientists have discovered and described a new supermassive dinosaur species with the most complete skeleton ever found of its type. At 85 feet long and weighing about 65 tons in life, Dreadnoughtus schrani is the largest land animal for which a body mass can be accurately calculated.

Its skeleton is exceptionally complete, with over 70 percent of the bones, excluding the head, represented. Because all previously discovered super-massive dinosaurs are known only from relatively fragmentary remains, Dreadnoughtus offers an unprecedented window into the anatomy and biomechanics of the largest animals to ever walk the Earth.

Dreadnoughtus schrani was astoundingly huge,” said Kenneth Lacovara, an associate professor in Drexel University’s College of Arts and Sciences, who discovered the Dreadnoughtus fossil skeleton in southern Patagonia in Argentina and led the excavation and analysis. “It weighed as much as a dozen African elephants or more than seven T. rex. Shockingly, skeletal evidence shows that when this 65-ton specimen died, it was not yet full grown. It is by far the best example we have of any of the most giant creatures to ever walk the planet.”

Lacovara and colleagues published the detailed description of their discovery, defining the genus and species Dreadnoughtus schrani, in the journal Scientific Reports from the Nature Publishing Group today. The new dinosaur belongs to a group of large plant eaters known as titanosaurs. The fossil was unearthed over four field seasons from 2005 through 2009 by Lacovara and a team including Lucio M. Ibiricu of the Centro Nacional Patagonico in Chubut, Argentina; the Carnegie Museum of Natural History’s Matthew Lamanna, and Jason Poole of the Academy of Natural Sciences of Drexel University, as well as many current and former Drexel students and other collaborators. These included three current NSF Graduate Research Fellows–current GRF Kristyn Voegele, and former GRFs Elena Schroeter and Paul Ullmann–all co-authors of this paper.

“The quality of this specimen has allowed us to study this new species in numerous aspects giving us closer to a holistic view than is possible for most dinosaur species,” said Voegele. “This could only be accomplished by collaborating with multiple experts–and without this collaboration our knowledge of this taxon would be fragmentary and not live up to the completeness and quality of the specimen. The NSF GRFP has enabled myself and two fellow collaborators to preform detailed analyses of this new species.”

“The fellowship awarded in 2013 acknowledged Kristyn’s scientific potential, and supports her contributions to this exciting discovery,” said Gisele Muller-Parker, program director for the Graduate Research Fellowship Program. “In addition to her research on dinosaur anatomy and biomechanics, Kristyn has been involved in a variety of related outreach activities, including an annual Community Dig Day and a Fossil Discovery Station for school visits at a fossil site in New Jersey.”

NSF funding also included an Earth Sciences award of the Geobiology and Low-Temperature Geochemistry program.

For more information on this research, please go to the Drexel press release.

-NSF-

Media Contacts

Maria C. Zacharias, NSF, (703) 292-8454, mzachari@nsf.gov

Rachel Ewing, Drexel University, (215) 895-2614, re39@drexel.edu

The National Science Foundation (NSF) is an independent federal agency that supports fundamental research and education across all fields of science and engineering. In fiscal year (FY) 2014, its budget is $7.2 billion. NSF funds reach all 50 states through grants to nearly 2,000 colleges, universities and other institutions. Each year, NSF receives about 50,000 competitive requests for funding, and makes about 11,500 new funding awards. NSF also awards about $593 million in professional and service contracts yearly.

 Get News Updates by Email 

Useful NSF Web Sites:

NSF Home Page: http://www.nsf.gov
NSF News: http://www.nsf.gov/news/
For the News Media: http://www.nsf.gov/news/newsroom.jsp
Science and Engineering Statistics: http://www.nsf.gov/statistics/
Awards Searches: http://www.nsf.gov/awardsearch/

 

11
Jun

West Nile Virus Notes

 

2014 budget is $30,097,170 than last year.  Failing to solve the problem means more money and resources (personnel) involved the next time there’s a spike in temperatures and drop in precipitation.

Current drought map

Drought monitor archive

CDC West Nile Virus stats (2002-2012) PDF archives

Mother Jones (2012) interactive maps

The middle class suburban areas appeared to support the appropriate combination of vegetation, open space, and potential vector habitat favoring WNV transmission. Wealthier neighborhoods had more vegetation, more diverse land use, and less habitat fragmentation likely resulting in higher biological diversity potentially protective against the WNV human transmission, e.g. the avian host “dilution effect” [45].

CDC WNV stats 2002-2012 by state

 

 TIME 2/28/2014

The biggest indicator of whether West Nile virus will occur is the maximum temperature of the warmest month of the year, which is why the virus has caused the most damage in hot southern states like Texas.

The UCLA model indicates that higher temperatures and lower precipitation will generally lead to more cases of West Nile

 

 

2012 Scientific American

A nearly frost-free winter followed by the summer’s drought has worsened the epidemic

 

West Nile Virus outbreak map

west nile virus