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Posts from the ‘Commentary’ Category

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.

For an arbitrary a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M230','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M230View MathML/a, choose a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M231','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M231View MathML/a such that a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M232','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M232View MathML/a in a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M233','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M233View MathML/a as a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M234','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M234View MathML/a. Then, for small enough ε,

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M235','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M235View MathML/a

(3.1)

Let a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M236','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M236View 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/M237','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M237View MathML/a for all a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M238','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M238View MathML/a, a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M239','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M239View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M240','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M240View MathML/a and a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M241','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M241View MathML/a for a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M242','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M242View MathML/a, and let

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)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M245','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M245View MathML/a

(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)

a onClick=popup('http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M252','MathML',630,470);return false; target=_blank href=http://www.boundaryvalueproblems.com/content/2015/1/4/mathml/M252View MathML/a

(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.

If the price decreases, we will simply charge the lower price.

Applicable discounts will be extended.

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
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Awards Searches: http://www.nsf.gov/awardsearch/

 

2
Mar

Turning Classrooms Into Video Games

2013-05-19 10.23.03

One thing that teachers continually explore is “how do I make my classroom even better for the students.”  Teaching can be a very frustrating but very rewarding profession, IF you are prepared to deal with change.  Students have many different learning styles, students have many different abilities, AND our society changes how we learn and what we prefer as time goes on.  You (as a teacher) also have your own style, and learning “what works for me as a teacher” is all part of the training process for teachers.

The preferred way of teaching when I was little was similar to the first teaching methods:  make the kid sit still and repeat, punish them if they don’t do it right.  It’s brutal, but the result of this kind of teaching is that my generation is one of the best educated generations ever.

And it’s not going to work in today’s society.

So teachers, like Paul Anderson, are trying new strategies that work better in today’s schools

His TED talk covers a method being tried by a number of teachers across the nation: turning the classroom into a video game by going from teacher-centric to student-centric types of learning.  It isn’t a “magic bullet”, though, and he’s careful to point this out.  It requires a lot of planning on the part of the teacher and an understanding that reading sophistication may be a barrier to some of it.  He doesn’t mention it, but another consequence can be loss of control in the classroom (and other teachers and administrators getting on you because your classroom is noisy and disrupts other teachers so that they can’t teach.)

If you’re interested in trying this, here’s some links to get you started:

Things to think about:
http://www.gamifeye.com/2012/10/21/education-and-training-from-game-based-learning-to-gamification/

Report from a teacher
http://www.eschoolnews.com/2013/03/12/how-i-turned-my-classroom-into-a-living-video-game-and-saw-achievement-soar/

Article on revitalizing education with games
http://www.eschoolnews.com/2013/02/12/how-game-mechanics-can-revitalize-education/

A game using Algebra.  (Okay, all you Algebra-terrified people… come play.  Yes, even if you’re an adult.)
http://labyrinth.thinkport.org/www/

Using gaming to engage girls (I can go along with this.  I’m female.  I game.  Works for me!)
http://www.eschoolnews.com/2013/01/17/how-to-engage-girls-with-gaming/ Gaming can be used to revitalize girls’ interests

16
Feb

Playing With Science — Surface Tension

Water impact (image courtesy Wikipedia commons)

In 1648, William Chillingworth wrote humorously about a religious dispute over “Whether a million of angels may not sit upon a needle’s point.” Although we can’t do experiments with angels or spirits, you can amuse yourself with informal investigations on surface tension and liquids using an eyedropper (or a something like a bottle for eye drops), a penny, and ordinary liquids, including water.

The volume of liquid that you can “pile up” in a single spot (like a penny) depends on the surface tension of the liquid — the way the liquid behaves when it comes into contact with the air (or another gas). The greater the surface tension, the more drops you can put on a penny before the liquid spills over the edge of the glass (or the surface of a penny.)

 

The Science Challenge:

Use a dropper to drop water onto the face of a penny, one drop at a time. How many drops will the penny hold before the water spills?  Run the test three times — what was the greatest number of drops of water you could put on the penny before surface tension broke and the water spilled?

Once you know that, it’s time to see how your water “shapes up” compared to other liquids (such as the bottled flavored teas I drink sometimes)

THINGS TO TEST:
Bottled water vs tap water
Tap water of different cities
Water vs milk
Water vs olive oil (or any other oil — the results may surprise you)
Water vs salt water
Water vs tea with sugar
Water vs detergent
Water vs soda
Water vs beer or wine
Soda vs beer or wine
Soda vs diet soda
Disinfectants vs water
… let your imagination run wild! Any liquid in any form can be tested against another liquid or against water.

Water is one of the stickiest substances around. If you put anything in water, the water will cling to it — in other words, it becomes wet. But sometimes we don’t WANT the water sticking to an object (like windshields or dishes) and in that case we turn to chemistry to look for a process or chemical that makes water less “sticky.”

Cleaning products like soaps that reduce the surface tension are among our most useful chemical compounds. To be a good surfactant (an acronym for SURface ACTive AgeNT), the chemical compound must have two parts on the molecule that react with water — a “water loving” (polar or ionic) and a water hating (hydrocarbon or fluorocarbon) part in the same molecule. These chemicals don’t combine with water to form a new compound but instead float on top of the water with the “water loving” part touching the water and the “water hating” part touching the air. The hydrocarbon or fluorocarbon parts of the surfactant interfere with the bonds between the water molecules at the surface of the water.

21
Dec

Pigging out at TRAC

I’ve been volunteering at Trinity River Audubon Center since 2009; long enough to be allowed to do “special projects” for them; projects that will start with research and end with publications in magazines and (hopefully) journals or conference presentations.

TRAC is 200 acres of a “blackfield remediation” site — an illegal dump that had polluted the Trinity and the neighborhood for over 30 years which was reclaimed and turned into an Audubon center.  A recent decision by the City of Dallas to turn a part of the Trinity River corridor area next to the center into a golf course has chased the feral hogs from there onto our property, with the result that we’re seeing a lot of landscape damage from these animals.

IMG_20131220_141155

 

This is a section of the trail near the building, where the hogs have been rooting all along the gravel walkways.  The damaged landscape left by their actions is vulnerable to erosion, and any native plants in this area that are destroyed are often replaced by invasives, which represents a step backwards in the effort to recreate a “pristine prairie environment” similar to what would have been here fifty or a hundred years ago.   The idea that you can take a damaged piece of land and magically return it to a pristine state is a bit of a pipe dream — we have been fighting a constant war with invasive species since the center’s opening.

So, this is my new research — find out about the hogs on THIS piece of property and see if we can manage them — because they are also destroying the juniper forest, as you can see in the two photos below.

IMG_20131220_141628

IMG_20131220_142132I’m going to focus on “manage them to minimize damage” rather than “eliminate” since much of the research indicates that they’re difficult to eradicate and that habitats free of hogs are just an invitation for other hogs to move in.

Here’s what I know:

* There are two to four different herds.  I’ve set up a cheap game camera (and am hoping it works) to start to get pictures of the pigs so we can identify them.

* the pigs are mainly going for areas with Johnson grass and areas with junipers (cedars).  There’s minor damage in other areas, but the Johnson grass places and juniper forest are the places that are most heavily damaged.

Here’s what I’m thinking:

* that it might be possible to landscape the area (brush piles and so forth) to make it less convenient habitat for the pigs.

* that when they tear up the invasive Johnson grass, they’re doing us a favor.  We can plant over those areas with native grasses and the hogs have done a lot of the removal and soil tilling work for us.

* would a maintained and controlled herd keep other herds from entering the land?

* Are there certain types of landscapes that the pigs don’t like?  In other words, do they avoid walking over cobblestone-sized rocks or do they avoid brush piles — what do they avoid and what do they prefer?

My research assistant, John Snodgrass, is hunting up web pages for me to look at on wild hogs, but so far the information seems to come down to: they breed rapidly, they’re destructive, trapping and killing are the best ways to get rid of them, and Wild Hogs Are Tasty.

So — I’m also soliciting thoughts and observations here — if you have a thought or an observation or an idea about hogs (remembering that this will be done by One Small Woman… so don’t advocate putting up 30 miles of barbed wire fence, ’cause it just ain’t happening), let me know and I’ll add it to my list.

Alternatively, if you have a game camera to loan me or want to help me come map the trails on the property with a GPS or help map the damage), let me know and you can be part of the team.

 

20
Nov

Hands-on STEM activities challenge students to define problems and determine solutions

It’s a system I’ve seen in eco-education that seems to be a growing trend in education — partnering with organizations to inspire and challenge students by giving them an opportunity to use math, science, and engineering skills to solve real-world problems.

A partnership between the Georgia Institute of Technology and the Griffin-Spalding County School system called “AMP-IT-UP”, is using a novel approach to encourage student creativity, and make these important courses come alive.

The new courses integrate basic science and math content with hands-on engineering design and construction. The idea is to get youngsters to think about engineering concepts by using math and science as they design and build projects — often for a specific “client.”  The project also monitors the students’ performance, collecting data to try to determine what the students learn, and whether the program is succeeding in engaging them.

For the AMP-IT-UP program, students will be challenged to defend their decisions and ideas with science and math.  The program plans to give them access to equipment such as 3-D printers, laser cutters and vinyl cutters.

“These classes build upon traditional classes where kids actually made things, such as making wooden boxes in what we used to call ‘wood shop,’ but with the addition of math, science and engineering design,” says Marion Usselman, an associate director for federal outreach and research of the Center for Education Integrating Science, Mathematics and Computing (CEISMC) at Georgia Tech and co-principal investigator and program director of the AMP-IT-UP.  “The emphasis in AMP-IT-UP is on students learning to define a problem–an engineering challenge–and then constructing a prototype and collecting data to find out whether the design works. They then change things based on the data.”

“What we are looking at is getting all students engaged in the act of making things, which allows them to have a better contextualization of math and science,” says Jeff Rosen, co-principal investigator for implementation and partnerships for AMP-IT-UP and a program director for robotics and engineering at CEISMC. “The whole idea of STEM (science, technology, engineering and mathematics) education is great, but the object is not to just do advanced math and science. It’s really about doing it all simultaneously, so you can get a true solid understanding of how everything works together.”

AMP-IT-UP is among the more than 100 currently active projects supported by NSF’s Math and Science Partnership (MSP), which has funded about 180 partnership-projects with local school districts since 2002.  For more on this project, check the link below.

20
Jul

Walking with Light

My new ebook, “Walking With Light” is available for download.  The revised version will be up in September.

It can be read online here: http://pyrts.pressbooks.com/

PDF download for reading is here:

Walking-With-Light-1374293171