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January 12, 2015

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

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