Abstract and Applied Analysis

Volume 2012 (2012), Article ID 368425, 14 pages

doi:10.1155/2012/368425

Research Article

Existence of Solutions for a Quasilinear Reaction Diffusion System

Canrong Tian

Department of Basic Sciences, Yancheng Institute of Technology, Jiangsu, Yancheng 224003, China

Received 26 February 2012; Revised 30 April 2012; Accepted 30 April 2012

Academic Editor: D. Anderson

Copyright © 2012 Canrong Tian. 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.

Abstract.

The degenerate reaction diffusion system has been applied to a variety of physical and engineering problems. This paper is extended the existence of solutions from the quasimonotone reaction functions (e.g., inhibitor-inhibitor mechanism) to the mixed quasimonotone reaction functions (e.g., activator-inhibitor mechanism). By Schauder fixed point theorem, it is shown that the system admits at least one positive solution if there exist a coupled of upper and lower solutions. This result is applied to a Lotka-Volterra predator-prey model.

1. Introduction

We consider a quasilinear reaction diffusion system in a bounded domain under coupled nonlinear boundary conditions. The system of equations is given in the form

where , is a bounded domain in with boundary , denotes the outward normal derivative on . It is assumed that the boundary is of class . It is also assumed that, for each , the functions , , and are Hölder continuous in . The density-dependent diffusion coefficient may have the property , which means that the elliptic operators are degenerate.

The quasilinear reaction diffusion system has been investigated extensively in the literature [1–3]. Recently by use of upper and lower solutions and its associated monotone iterations, [4, 5] deal with the scalar equation and the system endowed with the nonlinear Neumann-Robin boundary conditions, respectively. The paper in [6] is concerned with the existence, uniqueness, and asymptotic behavior for the quasilinear parabolic systems with the Dirichlet boundary condition. However, the requirement of the reaction functions in [4–6] are monotone nondecreasing. This paper relaxed the condition to mixed quasimonotone reaction functions, which leads to the difficult point that the ordered upper and lower solutions do not exist. To overcome it, we construct the coupled upper and lower solutions.

The purpose of this paper is to study the existence for the system (1.1) by the Schauder fixed point theorem. The rest of this paper is organized as follows. In Section 2 we show the existence by the method of upper and lower solutions and the Schauder fixed point theorem. An application is given in Section 3 to the Lotka-Volterra predator-prey model. The paper ends with Section 4 for some discussions.

2. Existence of Solutions

To the simplicity, throughout this paper, we denote

and let and be the respective space of -times differentiable and Hölder continuous functions in , where represents a domain or a section between two functions. For vector functions with -components we denote the above function space by and , respectively.

In this paper, we make the following hypothesis.

For each , the following conditions hold:

(i), and are in with , ;(ii) and for and ;(iii) are mixed quasimonotone -functions in .

In the above hypothesis, and are the sectors between a pair of coupled upper and lower solutions given by (2.8) below. It is allowed that for some and for a different . Particularly, if is a positive constant for all then system (1.1) becomes the standard coupled system of semilinear parabolic equations. Recall that a vector function is said to be mixed quasimonotone in if for each , there exist nonnegative integers and with such that the function is nondecreasing with respect to all component and is nonincreasing with respect to all component , where . Similarly, . Our approach to the existence problem is by the method of coupled upper and lower solutions which are defined as follows.

Definition 2.1. A pair of functions are called coupled upper and lower solutions of (1.1) if and if

Define

it follows from the following Hypothesis that

then the inverse exists and is an increasing function of . In view of

we may write (1.1) in the equivalent form

where . Thus the pair and , where and , satisfy the inequalities

is referred to as coupled upper and lower solutions of (2.6). For a given pair of coupled upper and lower solutions , we set

In Hypothesis - we allow which leads to a degenerate diffusion coefficient. If , we set , which ensures that has a positive lower bound. Since -, there exist smooth nonnegative functions , such that

In fact, it suffices to choose any , satisfying

Define for each ,

Since (2.9), and , and possess the property

Moreover, (2.6) is equivalent to

Thus the pair and , where and , satisfies the inequalities

are referred to coupled upper and lower solutions of (2.13).

The property (2.12) is quite useful for the construction of monotone convergent sequences. To ensure the existence of the sequence to be constructed in the iteration process (2.16) below we assume that either or for . Define a modified function by

Then by the above assumption, there exists such that for all .

By using and as the initial iteration we can construct sequences and from the nonlinear iteration process

The sequences and are well defined by the existence theorem of [1]. The following lemma gives the monotone property of these sequences.

Lemma 2.2. The sequences , governed by (2.16) possess the monotone property

Moreover, for each and are coupled upper and lower solutions of (1.1).

Proof. Let , . Then by (2.14) and (2.16), satisfies

Since by the mean value theorem,

for some intermediate value between and , we have

where

Since (2.14), the boundary and initial inequalities

In view of the definition of in (2.15), the function of (2.20) is bounded. From the weak maximum principle, it follows on . This gives and thus . A similar argument yields and .

Moreover, letting , by (2.12), (2.16), and after the similar above argument

where

for some intermediate value between and   . It follows again from the weak maximum principle that and thus . The above conclusions show that

Now we show that and are coupled upper and lower solutions of (1.1). Since (2.25), for . It suffices to show that and satisfy (2.14). Since (2.12) and (2.16), we have

Next we use an induction method. We assume that and are coupled upper and lower solutions of (1.1) and satisfying the following relation:

Then by choosing and as the coupled upper and lower solutions and , after the similar above argument, we have

and are coupled upper and lower solutions of (1.1). The conclusion of the lemma follows from the induction principle.

Theorem 2.3. Let , be a pair of coupled upper and lower solutions of (1.1), and let hypothesis hold. Assume that either for some or . Then the problem (1.1) has at least one solution .

Proof. We first consider the problem (2.13), where is replaced by . For each , we define operators and by

where

Define also

Then the system (2.13), in which is replaced by , may be written in the form

where and are given in (2.11). Given any and any , we consider the scalar problem

It follows from the existence theorem of [1] (Chapter V, Section 7) that (2.33) has a unique solution . In fact, the inverse exists and is a positive compact operator on . This implies that the equation

has a unique solution . Let be the closed bounded convex sunset given by

By the compact property on and the hypothesis on the operator is compact on . We show that maps to itself.

Let be given, and . After the similar argument of the proof of Lemma 2.2, we conclude , therefore maps to itself. It follows from the Schauder fixed point theorem that (2.13) with being replaced by has at least one solution . Since , it follows from (2.15) that for . Thus is also the solution of (2.13). Therefore the existence of the solution to (1.1) is proved.

3. Applications

As an application of the results obtained in the previous section we consider a Lotka-Volterra predator model. This model involves two species and that are governed by the system

where are positive constants , the initial functions for have a positive lower bound. The density-dependent diffusion coefficients .

It is easy to verify that if and satisfy and the following inequalities:

then the pair are coupled upper and lower solutions of (3.1).

To guarantee (3.2), we seek such a pair in the form

where for each ,   and are positive constants to be chosen, is the inverse of (2.3), and is the (normalized) positive eigenfunction corresponding to the smallest eigenvalue of the eigenvalue problem

The constant will be determined in the following discussion. If we set

then the first and second inequalities of (3.2) are satisfied. The third and fourth inequalities become

By (3.4) and , the above inequalities are satisfied by some sufficiently small if

Since , by L’Hopital’s rule,

we see that there exists such that the inequalities in (3.7) are satisfied by every if we impose the condition

By (3.3), the fifth inequalities of (3.2) are trivially satisfied, and the sixth inequalities of (3.2) become

Substituting (3.4) into (3.10) yields

It is obvious that the above relations hold for any if . In the general case the relations and , where is defined in (2.3), implies that (3.11) is satisfied if

Since ,

If we impose the condition

then by setting

(3.12) is satisfied. If the below (3.16) holds, then (3.5) and (3.9) are satisfied. Thus all inequalities of (3.2) are satisfied. Directly applying Theorem 2.3, we have the following theorem.

Theorem 3.1. Suppose the initial functions in (2.8) for . Let

and let and satisfy - with . Assume that either or

for some constants . Then the system (3.1) admits at least one positive solution.

Remark 3.2. Pao and Ruan [5] have considered a Lotka-Volterra competition model with density-dependent diffusion, where the coefficient of the system (3.1) is negative. The difference between them is that our method does not require that the reaction functions possess the monotone nondecreasing property. The condition for the existence for the solutions of the competition model is , while the condition for the existence for the solutions of the predator model is .

Remark 3.3. In a special case , for , (3.1) becomes

Then the condition (3.17) is trivially satisfied. The conclusions in Theorem 3.1 hold true for (3.18). In fact, if , the condition (3.17) is also trivial true, hence Theorem 2.3 is also valid. After the similar proof as Theorem 3.1, we conclude that Theorem 3.1 holds true for semilinear parabolic system.

4. Discussions

The intension of the present paper is to demonstrate the existence of solutions for the degenerate diffusion reaction system with nonlinear boundary condition. Our method is to look for the positive solution by constructing the coupled upper and lower solutions. The virtue of the technique is that it helps to extend the results for the scalar equation to the coupled system. Our existence theorem of Theorem 2.3 in this paper is applicable to various Lotka-Volterra models, such as competition, predator-prey, or mutualism model, while the method in [6] is not applicable to predator-prey model.

Since Levin and Segel illuminated the important role of the diffusion on the patterns in [7], a number of Lotka-Volterra models with constant diffusion have been investigated in the past three decades. In fact the concern of the density-dependent diffusion is also reasonable in animal disperse model (see [8] for a review). Our study is a starting attempt to consider the role of the density-dependent diffusion on Lotka-Volterra model. In biological terms, the results of Theorems 3.1 imply that if the rate of intraspecific competition of the predator is large, the two species are coexistent. The results also have applicability to 3 species model. Note that for Lotka-Volterra predator-prey model with constant diffusion, when the rate of intraspecific competition of the prey is large, the two species are both extinct. When the density-dependent diffusion is taken into account, it is an open problem whether there exist the extinct phenomena.

Acknowledgment

The work is partially supported by PRC Grant NSFC 10801115.

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