The Existence and Behavior of Solutions for Nonlocal Boundary Problems

  • Yuandi Wang1Email author and

    Affiliated with

    • Shengzhou Zheng2

      Affiliated with

      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):

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

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq2_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq4_HTML.gif ,

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

      and http://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

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

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

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

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

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq27_HTML.gif and http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq29_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq30_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq31_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq32_HTML.gif .

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

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

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

      (v) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq43_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq44_HTML.gif .

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

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

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

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

      (viii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq61_HTML.gif ( http://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

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

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

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

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

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

      has a unique solution http://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

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

      Exactly, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq78_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq79_HTML.gif , where http://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 http://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   http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq83_HTML.gif function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq85_HTML.gif , satisfying
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ11_HTML.gif
      (2.11)

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

      Set

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

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

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

      (H3) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq99_HTML.gif for some http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq102_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq103_HTML.gif .

      Proof.

      Recall that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ14_HTML.gif
      (2.14)
      The Nemytskii operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq104_HTML.gif is defined as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq105_HTML.gif . The fact
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq107_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq108_HTML.gif for some http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ16_HTML.gif
      (2.16)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq110_HTML.gif . So, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq111_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq112_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ17_HTML.gif
      (2.17)
      From http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq113_HTML.gif , we infer that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ18_HTML.gif
      (2.18)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq114_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq115_HTML.gif , we can choose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq116_HTML.gif such that
      http://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

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ20_HTML.gif
      (2.20)
      Due to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq118_HTML.gif , Hölder inequality follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ21_HTML.gif
      (2.21)
      The hypothesis of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq119_HTML.gif means that one can find an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq120_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ22_HTML.gif
      (2.22)
      Obviously, if http://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
      http://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
      http://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

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

      (Hr2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq123_HTML.gif   is locally Lipschitz continuous and   http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq125_HTML.gif , has a unique http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq126_HTML.gif for some http://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:
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq128_HTML.gif is independent of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq130_HTML.gif and http://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+) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq132_HTML.gif   is nondecreasing with   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq133_HTML.gif , and

      http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq134_HTML.gif ) be satisfied. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq135_HTML.gif is nonnegative, then the solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq137_HTML.gif . Multiplying the equation with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq138_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq139_HTML.gif , we have
      http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq140_HTML.gif ), we can get following equality: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq141_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ29_HTML.gif
      (3.5)
      So,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq142_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq143_HTML.gif and the restriction http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq144_HTML.gif are used. Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ31_HTML.gif
      (3.7)

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

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

      We denote by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq149_HTML.gif such that the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq150_HTML.gif of problem (3.1) satisfies
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ32_HTML.gif
      (3.8)

      Proof.

      Multiplying by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq151_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq152_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ33_HTML.gif
      (3.9)
      That is,
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ35_HTML.gif
      (3.11)
      Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ36_HTML.gif
      (3.12)
      By using the embedding http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq153_HTML.gif and letting http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ37_HTML.gif
      (3.13)

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

      Theorem 3.4.

      If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq156_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq157_HTML.gif , then the solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq159_HTML.gif -norm for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq161_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq162_HTML.gif . By multiplying with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq163_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq164_HTML.gif ) and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq165_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ38_HTML.gif
      (3.14)
      That is,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ39_HTML.gif
      (3.15)
      But,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ40_HTML.gif
      (3.16)
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq166_HTML.gif Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ41_HTML.gif
      (3.17)
      where Young's inequality, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq168_HTML.gif again with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq169_HTML.gif and choose http://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:
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ43_HTML.gif
      (3.19)
      Set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq171_HTML.gif , then we deduce that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ44_HTML.gif
      (3.20)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq172_HTML.gif the inequality (3.20) implies
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq173_HTML.gif -norm of the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq174_HTML.gif , that is, http://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 http://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:

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

      The function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq179_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq180_HTML.gif with respect to http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq183_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq184_HTML.gif ;

      (Hn2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq185_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq186_HTML.gif satisfies the Carathéodory condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq187_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq189_HTML.gif , with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq191_HTML.gif for some http://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

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

      and

      (Hk)The function   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq193_HTML.gif   satisfies the Carathéodory condition on   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq194_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq195_HTML.gif   and f http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq196_HTML.gif with
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq197_HTML.gif , has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq198_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq199_HTML.gif .

      Proof.

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

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

      That is, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq205_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq207_HTML.gif .

      4.1. http://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   http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq209_HTML.gif   such that

      http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq211_HTML.gif and then integrate over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq212_HTML.gif , and we find that
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ57_HTML.gif
      (4.12)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq213_HTML.gif for http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq215_HTML.gif small enough ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq216_HTML.gif ); it is not difficult to get
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ59_HTML.gif
      (4.14)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq217_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq218_HTML.gif . Therefore, by multiplying with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq219_HTML.gif and integrating over http://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. http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq222_HTML.gif , then the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq223_HTML.gif of problem (4.9) satisfies
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq224_HTML.gif and integrate over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq225_HTML.gif , then we reach that
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ62_HTML.gif
      (4.17)
      Hence,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq226_HTML.gif , so,
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ64_HTML.gif
      (4.19)

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

      Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq229_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq230_HTML.gif small enough, then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ65_HTML.gif
      (4.20)
      Multiplying with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq231_HTML.gif , then integrating over http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq232_HTML.gif , we obtain that
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq233_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq234_HTML.gif   ( http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq235_HTML.gif ) such that

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

      for all http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq238_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq239_HTML.gif decay to zero as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq240_HTML.gif for some small functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq241_HTML.gif .

      Proof.

      We use http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq243_HTML.gif . Thus, we get that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq244_HTML.gif as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_Equ70_HTML.gif
      (4.25)
      then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq245_HTML.gif also decay to zero (as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F484879/MediaObjects/13661_2008_Article_848_IEq246_HTML.gif ) if http://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

      References

      1. Amann H: Quasilinear parabolic problems via maximal regularity. Advances in Differential Equations 2005, 10(10):1081–1110.MATHMathSciNet
      2. Amosov AA: Global solvability of a nonlinear nonstationary problem with a nonlocal boundary condition of radiative heat transfer type. Differential Equations 2005, 41(1):96–109. 10.1007/s10625-005-0139-9MATHMathSciNetView Article
      3. Day WA: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quarterly of Applied Mathematics 1983, 40(4):468–475.MATH
      4. Friedman A: Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions. Quarterly of Applied Mathematics 1986, 44(3):401–407.MATHMathSciNet
      5. Muraveĭ LA, Filinovskiĭ AV: On a problem with nonlocal boundary condition for a parabolic equation. Mathematics of the USSR-Sbornik 1993, 74(1):219–249. 10.1070/SM1993v074n01ABEH003345MathSciNetView Article
      6. Feller W: The parabolic differential equations and the associated semi-groups of transformations. Annals of Mathematics 1952, 55(3):468–519. 10.2307/1969644MATHMathSciNetView Article
      7. Lions JL, Magenes E: Non-Homogeneous Boundary Value Problems and Applications. Vol. I. Springer, Berlin, Germany; 1972:xvi+357.MATHView Article
      8. Lions JL, Magenes E: Non-Homogeneous Boundary Value Problems and Applications. Vol. II. Springer, Berlin, Germany; 1972:xi+242.MATHView Article
      9. Lasiecka I: Unified theory for abstract parabolic boundary problems—a semigroup approach. Applied Mathematics and Optimization 1980, 6(1):287–333. 10.1007/BF01442900MATHMathSciNetView Article
      10. Amann H: Feedback stabilization of linear and semilinear parabolic systems. In Semigroup Theory and Applications (Trieste, 1987), Lecture Notes in Pure and Applied Mathematics. Volume 116. Edited by: Clement P, Invernizzi S, Mitidieri E, Vrabie II. Dekker, New York, NY, USA; 1989:21–57.
      11. Agarwal RP, Bohner M, Shakhmurov VB: Linear and nonlinear nonlocal boundary value problems for differential-operator equations. Applicable Analysis 2006, 85(6–7):701–716. 10.1080/00036810500533153MATHMathSciNetView Article
      12. Ashyralyev A: Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces. Journal of Evolution Equations 2006, 6(1):1–28. 10.1007/s00028-005-0194-yMATHMathSciNetView Article
      13. Capasso V, Kunisch K: A reaction-diffusion system modelling man-environment epidemics. Annals of Differential Equations 1985, 1(1):1–12.MATHMathSciNet
      14. Taira K: Diffusion Processes and Partial Differential Equations. Academic Press, Boston, Mass, USA; 1988:xviii+452.MATH
      15. Carl S, Heikkilä S: Discontinuous reaction-diffusion equations under discontinuous and nonlocal flux conditions. Mathematical and Computer Modelling 2000, 32(11–13):1333–1344.MATHMathSciNetView Article

      Copyright

      © Y. Wang and S. Zheng. 2009

      This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.