Open Access

The Existence and Behavior of Solutions for Nonlocal Boundary Problems

Boundary Value Problems20092009:484879

DOI: 10.1155/2009/484879

Received: 16 October 2008

Accepted: 23 March 2009

Published: 31 March 2009

Abstract

The purpose of this work is to investigate the uniqueness and existence of local solutions for the boundary value problem of a quasilinear parabolic equation. The result is obtained via the abstract theory of maximal regularity. Applications are given to some model problems in nonstationary radiative heat transfer and reaction-diffusion equation with nonlocal boundary flux conditions.

1. Introduction

The existence of solutions for quasilinear parabolic equation with boundary conditions and initial conditions can be discussed by maximal regularity, and more and more works on this field show that the maximal regularity method is efficient. Here we will use some of recently results developed by H. Amann to investigate a specific boundary value problems and then apply the existence theorem to two nonlocal problems.

This paper consists of three parts. In the next section we present and prove the existence and unique theorem of an abstract boundary problem. Then we give some applications of the results in Sections 3 and 4 to two reaction-diffusion model problems that arise from nonstationary radiative heat transfer in a system of moving absolutely black bodies and a reaction-diffusion equation with nonlocal boundary flux conditions.

2. Notations and Abstract Result

We consider the following quasilinear parabolic initial boundary value problem (IBVP for short):

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ1_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq1_HTML.gif is a bounded strictly Lipschitz domain with its boundary https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq4_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ2_HTML.gif
(2.2)

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq5_HTML.gif is a second-order strongly elliptic differential operator with the boundary operator given by

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ3_HTML.gif
(2.3)

The coefficient matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq6_HTML.gif satisfies regularity conditions on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq7_HTML.gif , respectively. The directional derivative https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq9_HTML.gif is the outer unit-normal vector on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq10_HTML.gif ; the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq11_HTML.gif is defined as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq12_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq13_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq14_HTML.gif denotes the trace operator.

We introduce precise assumptions:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ4_HTML.gif
(2.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq15_HTML.gif ) are Carathéodory functions; that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq16_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq17_HTML.gif ) is measurable in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq18_HTML.gif (resp., in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq19_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq20_HTML.gif and continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq21_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq22_HTML.gif (resp., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq23_HTML.gif . More general, the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq24_HTML.gif can be a nonlocal function, for example, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq25_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq26_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq28_HTML.gif be Banach spaces, we introduce some notations as follows:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq30_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq32_HTML.gif .

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq33_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq35_HTML.gif .

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq36_HTML.gif all continuous linear operators from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq37_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq38_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq39_HTML.gif .

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq40_HTML.gif denotes the Nemytskii operator induced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq41_HTML.gif .

(v) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq42_HTML.gif denotes the set of all locally Lipschitz-continuous functions from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq43_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq44_HTML.gif .

(vi) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq46_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq47_HTML.gif , denotes the set of all Carathéodory functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq48_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq49_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq50_HTML.gif , and there exists a nondecreasing function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq51_HTML.gif with

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ5_HTML.gif
(2.5)

Particularly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq52_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq53_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq54_HTML.gif .

(vii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq55_HTML.gif denotes the Sobolev-Slobodeckii space for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq57_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq58_HTML.gif , especially, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq59_HTML.gif ; and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ6_HTML.gif
(2.6)

(viii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq61_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq62_HTML.gif is the set of integral numbers), is defined as

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ7_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq63_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq64_HTML.gif is the dual space of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq65_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq66_HTML.gif is the formally adjoint operator.

(x) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq67_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq69_HTML.gif is an interval in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq70_HTML.gif .

(xi) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq71_HTML.gif denotes all maps https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq72_HTML.gif possessing the property of maximal   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq73_HTML.gif   regularity on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq74_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq75_HTML.gif , that is, given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq76_HTML.gif , the initial problem

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ8_HTML.gif
(2.8)

has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq77_HTML.gif .

Now we turn to discuss the local existence result. We write

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ9_HTML.gif
(2.9)
then,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ10_HTML.gif
(2.10)

Exactly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq78_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq79_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq80_HTML.gif denotes the Banach space of all functions being bounded and uniformly continuous in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq81_HTML.gif . So, we will not emphasize it in the following.

A (weak)  solution   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq82_HTML.gif of IBVP (2.1) is defined as a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq83_HTML.gif function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq85_HTML.gif , satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ11_HTML.gif
(2.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq86_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq87_HTML.gif denote the obvious duality pairings on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq89_HTML.gif , respectively.

Set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ12_HTML.gif
(2.12)

After these preparations we introduce the following hypotheses:

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq91_HTML.gif .

(H2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq92_HTML.gif   with   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq93_HTML.gif , and there exists a   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq94_HTML.gif   such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ13_HTML.gif
(2.13)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq95_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq96_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq97_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq98_HTML.gif .

(H3) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq99_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq100_HTML.gif .

Theorem 2.1.

Let assumptions (H1)-(H3) be satisfied. Then for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq101_HTML.gif the quasilinear problem (2.1) possesses a unique weak solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq102_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq103_HTML.gif .

Proof.

Recall that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ14_HTML.gif
(2.14)
The Nemytskii operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq104_HTML.gif is defined as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq105_HTML.gif . The fact
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ15_HTML.gif
(2.15)

shows the maximal regularity of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq106_HTML.gif . By [1, Theorem  2.1], if, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq107_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq108_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq109_HTML.gif , then the existence and the uniqueness of a local solution will be proved.

The remain work is to check the Lipschitz-continuity. Set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ16_HTML.gif
(2.16)
Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq110_HTML.gif . So, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq111_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq112_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ17_HTML.gif
(2.17)
From https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq113_HTML.gif , we infer that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ18_HTML.gif
(2.18)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq114_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq115_HTML.gif , we can choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq116_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ19_HTML.gif
(2.19)

On the other hand, the hypotheses guarantee that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ20_HTML.gif
(2.20)
Due to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq118_HTML.gif , Hölder inequality follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ21_HTML.gif
(2.21)
The hypothesis of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq119_HTML.gif means that one can find an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq120_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ22_HTML.gif
(2.22)
Obviously, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq121_HTML.gif , the above inequality is followed from (2.20) immediately. Hence it follows from (2.19) and (2.22) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ23_HTML.gif
(2.23)

This ends the proof.

We apply the above theorem to the following two examples in next sections. For this, in the remainder we suppose that hypotheses (H1)-(H2) hold and that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ24_HTML.gif
(2.24)

3. A Radiative Heat Transfer Problem

We see a nonlinear initial-boundary value problem, which, in particular, describes a nonstationary radiative heat transfer in a system of absolutely black bodies (e.g., refer to [2]). A problem is

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ25_HTML.gif
(3.1)

3.1. Local Solvability

We assume that (Hr)

(Hr1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq122_HTML.gif ;

(Hr2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq123_HTML.gif   is locally Lipschitz continuous and   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq124_HTML.gif .

Theorem 3.1.

Let assumptions (H1)-(H2) and (Hr) be satisfied. Then problem (3.1), for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq125_HTML.gif , has a unique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq126_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq127_HTML.gif .

Proof.

Note that the embedding (2.14) holds:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ26_HTML.gif
(3.2)

Hence Theorem 2.1 implies the result immediately.

In fact, Amosov proved in 2005 the uniqueness of the solution for a simple case, that is, problem in which the matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq128_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq129_HTML.gif (see [2, Theorem  1.4]). In this paper, we also can get the positivity of the solution and the estimates of the solution in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq131_HTML.gif in this part. We have tried to achieve the global existence, but it is still an open problem.

In the rest of this section, we always assume that (H1)-(H2) and (Hr) hold.

3.2. Positivity

Assume that

(H+) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq132_HTML.gif   is nondecreasing with   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq133_HTML.gif , and

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ27_HTML.gif
(3.3)

Theorem 3.2.

Let assumption ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq134_HTML.gif ) be satisfied. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq135_HTML.gif is nonnegative, then the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq136_HTML.gif of problem (3.1) is also nonnegative.

Proof.

Put https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq137_HTML.gif . Multiplying the equation with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq138_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq139_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ28_HTML.gif
(3.4)
By using the assumption of ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq140_HTML.gif ), we can get following equality: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq141_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ29_HTML.gif
(3.5)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ30_HTML.gif
(3.6)
At the last inequality, the monotonity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq142_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq143_HTML.gif and the restriction https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq144_HTML.gif are used. Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ31_HTML.gif
(3.7)

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq145_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq146_HTML.gif . The assertion follows.

3.3. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq147_HTML.gif -norm

We denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq148_HTML.gif the maximal interval of the solution of problem (3.1).

Lemma 3.3.

There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq149_HTML.gif such that the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq150_HTML.gif of problem (3.1) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ32_HTML.gif
(3.8)

Proof.

Multiplying by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq151_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq152_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ33_HTML.gif
(3.9)
That is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ34_HTML.gif
(3.10)
As similar as the inequality (3.6), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ35_HTML.gif
(3.11)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ36_HTML.gif
(3.12)
By using the embedding https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq153_HTML.gif and letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq154_HTML.gif small enough, it is easy to get that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ37_HTML.gif
(3.13)

3.4. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq155_HTML.gif -norm

Theorem 3.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq156_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq157_HTML.gif , then the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq158_HTML.gif of problem (3.1) is bounded with its https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq159_HTML.gif -norm for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq160_HTML.gif .

Proof.

From the hypothesis (H1) and embedding (2.10), one has that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq162_HTML.gif . By multiplying with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq164_HTML.gif ) and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq165_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ38_HTML.gif
(3.14)
That is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ39_HTML.gif
(3.15)
But,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ40_HTML.gif
(3.16)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq166_HTML.gif Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ41_HTML.gif
(3.17)
where Young's inequality, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq167_HTML.gif , has been used at the last inequality. We apply the embedding https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq168_HTML.gif again with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq169_HTML.gif and choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq170_HTML.gif small enough, then we attain the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ42_HTML.gif
(3.18)
By Gronwall's inequality, the inequality (3.18) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ43_HTML.gif
(3.19)
Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq171_HTML.gif , then we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ44_HTML.gif
(3.20)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq172_HTML.gif the inequality (3.20) implies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ45_HTML.gif
(3.21)

The claim follows.

One immediate consequence of the above theorem is.

Corollary 3.5.

The https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq173_HTML.gif -norm of the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq174_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq175_HTML.gif , of problem (3.1) is nonincreasing if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq176_HTML.gif .

4. A Nonlocal Boundary Value Problem

We now consider the problem (2.1) with the following boundary value condition:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ46_HTML.gif
(4.1)

The function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq177_HTML.gif in (4.1) can be in nonlocal form.

IBVP (2.1) with a nonlocal term stands, for example, for a model problem arising from quasistatic thermoelasticity. Results on linear problems can be found in [35]. As far as we know, this kind of nonlocal boundary condition appeared first in 1952 in a paper [6] by W. Feller who discussed the existence of semigroups. There are other problems leading to this boundary condition, for example, control theory (see [712] etc.). Some other fields such as environmental science [13] and chemical diffusion [14] also give rise to such kinds of problems. We do not give further comments here.

Carl and Heikkilä [15] proved the existence of local solutions of the semilinear problem by using upper and lower solutions and pseudomonotone operators. But their results based on the monotonicity hypotheses of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq179_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq180_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq181_HTML.gif .

In this section, we assume that (H1) and (H2) always hold and assume that

(Hn1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq183_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq184_HTML.gif ;

(Hn2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq185_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq186_HTML.gif satisfies the Carathéodory condition on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq187_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq188_HTML.gif .

By the embedding theorem and Theorem 2.1, we get immediately.

Theorem 4.1.

Suppose hypotheses of (Hn) satisfy. Then problem (2.1), for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq189_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq190_HTML.gif defined in (4.1) has a unique https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq191_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq192_HTML.gif .

For the simplicity in expression, we turn to consider a problem with nonlocal boundary value

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ47_HTML.gif
(4.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ48_HTML.gif
(4.3)

and

(Hk)The function   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq193_HTML.gif   satisfies the Carathéodory condition on   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq195_HTML.gif   and f https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq196_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ49_HTML.gif
(4.4)

Theorem 4.2.

Let assumption (Hk) be satisfied. Then Problem (4.2), for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq197_HTML.gif , has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq198_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq199_HTML.gif .

Proof.

First, we see that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ50_HTML.gif
(4.5)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq200_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq201_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq202_HTML.gif . Consequently, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq203_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ51_HTML.gif
(4.6)
Similarly, from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq204_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ52_HTML.gif
(4.7)
Combining two inequalities (4.6) and (4.7), we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ53_HTML.gif
(4.8)

The claim follows immediately from Theorem 4.1.

A special case of problem (4.2) is

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ54_HTML.gif
(4.9)

That is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq205_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq206_HTML.gif in (4.9) are independent of gradient https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq207_HTML.gif .

4.1. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq208_HTML.gif -norm

In order to discuss the global existence of solution, in the rest of this section we assume the following.

(Hkl)Suppose there exists a continuous function   https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq209_HTML.gif   such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ55_HTML.gif
(4.10)

Lemma 4.3.

There exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq210_HTML.gif such that the solution of problem (4.9) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ56_HTML.gif
(4.11)

Proof.

We multiply the first equation in (4.9) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq211_HTML.gif and then integrate over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq212_HTML.gif , and we find that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ57_HTML.gif
(4.12)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq213_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq214_HTML.gif , by interpolation inequality and Young's inequality we have that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ58_HTML.gif
(4.13)
Apply Young's inequality again and then choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq215_HTML.gif small enough ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq216_HTML.gif ); it is not difficult to get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ59_HTML.gif
(4.14)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq217_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq218_HTML.gif . Therefore, by multiplying with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq219_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq220_HTML.gif , the inequality (4.14) follows the claim.

4.2. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq221_HTML.gif -norm

Lemma 4.4.

Let assumptions of Lemma 4.3 be satisfied. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq222_HTML.gif , then the solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq223_HTML.gif of problem (4.9) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ60_HTML.gif
(4.15)

Proof.

We multiply the first equation in (4.9) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq224_HTML.gif and integrate over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq225_HTML.gif , then we reach that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ61_HTML.gif
(4.16)
As the same as the inequality (4.13), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ62_HTML.gif
(4.17)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ63_HTML.gif
(4.18)
We might as well assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq226_HTML.gif , so,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ64_HTML.gif
(4.19)

The boundedness of solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq227_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq228_HTML.gif is used in above deduction.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq229_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq230_HTML.gif small enough, then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ65_HTML.gif
(4.20)
Multiplying with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq231_HTML.gif , then integrating over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq232_HTML.gif , we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ66_HTML.gif
(4.21)
By a similar limitation process as in (3.21), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ67_HTML.gif
(4.22)

This closes the end of proof.

4.3. Decay Behavior

In order to investigate the decay behavior of solution for problem (4.9), we assume that

(Hkd) there are two continuous function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq233_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq234_HTML.gif   ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq235_HTML.gif ) such that

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ68_HTML.gif
(4.23)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq236_HTML.gif .

Theorem 4.5.

Let the assumption (Hkd) be satisfied and, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq237_HTML.gif be the solution of problem (4.9) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq238_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq239_HTML.gif decay to zero as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq240_HTML.gif for some small functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq241_HTML.gif .

Proof.

We use https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq242_HTML.gif to multiply the first equation in the system (4.9) and then integrate over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq243_HTML.gif . Thus, we get that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ69_HTML.gif
(4.24)
In the above process the inequality (4.13) is used. If we choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq244_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ70_HTML.gif
(4.25)
then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ71_HTML.gif
(4.26)

This ends the proof.

Moreover, one can verify that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq245_HTML.gif also decay to zero (as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq246_HTML.gif ) if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq247_HTML.gif

Declarations

Acknowledgments

The first author wishes to thank Professor Herbert Amann for many useful discussions concerning the problem of this paper. The author also want to thank the referees' suggestions. This work is supported partly by the National NSF of China (Grant nos. 10572080 and 10671118) and by Shanghai Leading Academic Discipline Project (no. J50101).

Authors’ Affiliations

(1)
Department of Mathematics, Shanghai University
(2)
Department of Mathematics, Beijing Jiaotong University

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Copyright

© Y. Wang and S. Zheng. 2009

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