Periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions

In this article, we study the periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions. By using the theory of Leray-Schauder degree, we obtain the existence of a nontrivial nonnegative time periodic solution.

The aim of this work is to consider the following periodic problem for a quasilinear parabolic equation:

where Ω is a bounded domain in with smooth boundary ∂Ω, denotes the outward normal derivative on ∂Ω, , satisfies some suitable smoothness and structure conditions. This model can be used to describe the models for some interesting phenomena in mathematical biology, fisheries and wildlife management. The function gives the number of individuals (per unit area) of the species at position x and time t, where x represents the spatial variable and t represents the time. The term models a tendency to avoid high density in the habitat, describes the ways in which a given population grows and shrinks over time, as controlled by birth, death, emigration or immigration, and the Neumann boundary condition models the trend of the biology population who survive on the boundary.

In last decades, linear parabolic equations with nonlocal terms have been investigated by numerous researchers [1-4]. A typical model was submitted by Allegretto and Nistri [1] and they proposed the following equation:

with the Dirichlet boundary conditions. Also, according to the actual needs, many authors divert attention to nonlinear diffusion equations with nonlocal terms such as the porous equation [5,6] with a typical form

and the p-Laplacian equation [7] with a typical form

Equation (1.4) is degenerate if and singular if . Equation (1.5) is degenerate if and singular if . Only the cases and are considered with a few exceptions. All these equations are considered with the Dirichlet boundary condition which describes that the boundary is lethal to the species. Moreover, the methods in these papers are all based on the theory of Leray-Schauder degree. However, results on the quasilinear periodic parabolic equations with nonlocal terms and Neumann boundary conditions are few. In a recent paper [8], Wang and Yin considered the following periodic Neumann boundary value problem:

where . By the parabolic regularized method and the theory of Leray-Schauder degree, they established the existence of nontrivial nonnegative periodic solutions.

Inspired by the work of [8], we consider the periodic solutions of the Neumann boundary value problem of a quasilinear parabolic equation with nonlocal terms. Compared with the Dirichlet boundary condition, the Neumann boundary condition causes an additional difficulty in establishing some a priori estimates. On the other hand, different from the cases of the Dirichlet boundary condition, an auxiliary problem for (1.1)-(1.3) is considered for using the theory of Leray-Schauder degree. We prove that this problem (1.1)-(1.3) admits a nontrivial nonnegative periodic solution, that is, the following theorem.

Theorem 1If assumptions (A1), (A2), (A3) hold, then problem (1.1)-(3.3) admits a nontrivial nonnegative periodic solution.

The article is organized in the following way. In Section 2, we give some necessary preliminaries including the auxiliary problem. In Section 3, we establish some necessary a priori estimations of the solutions of the auxiliary problem. Then we show the proof of the main result of this paper.

In the paper, we assume that

(A1) and there exist two constants such that

where is the class of functions which are continuous in and T-periodic with respect to t. Furthermore, is continuous with respect to u.

(A2) is a bounded continuous functional satisfying

where C is a positive constant independent of u, , .

(A3) and satisfies that

where is the first eigenvalue of the Laplacian equation on ω with zero boundary and is the corresponding eigenfunction.

Since the regularity follows from a quite standard approach, we focus on the discussion of weak solutions in the following sense.

Definition 1 A function u is said to be a weak solution of problem (1.1)-(1.3), if and satisfies

for any with .

In order to use the theory of Leray-Schauder degree, we introduce a map by considering the following auxiliary problem:

where ε is a sufficiently small positive constant, is a parameter and . Then we can define a map with . Applying classical estimates (see [9]), we can see that is bounded by , and is Hölder continuous in . Then, by the Arzela-Ascoli theorem, the map G is compact. So, the map G is a compact continuous map. Let , where , we can see that the nonnegative solution of problem (2.2)-(2.4) is also a nonnegative fixed point of the map . So, we will study the existence of nonnegative fixed points of the map instead of a nonnegative solution of problem (2.2)-(2.4). And the desired solution u of (1.1)-(1.3) would be obtained as a limit point of .

First, by the same method as in [4], we can obtain the nonnegativity of the solutions of problem (2.2)-(2.4).

Lemma 1If a nontrivial functionsolves, then

In the following, we will show some a priori estimates for the upper bound of a nonnegative periodic solution of problem (2.2)-(2.4). Here and below, we denote by () the norm.

Lemma 2For, letbe a nonnegative periodic solution which solves, then there exists a constantKindependent ofλ, εsuch that

where.

Proof Multiplying (2.2) by () and integrating over Ω, we have

and hence

where () are positive constants independent of u and m. Assume that and set

then . For convenience, we denote by C a positive constant independent of k and m, which may take different values. From (3.2), we obtain

By using the Gagliardo-Nirenberg inequality, we have

with

By inequalities (3.3), (3.4) and the fact that , we obtain the following differential inequality:

Let

we have

By Young’s inequality,

where , , , , and . Set

then we obtain

Here we have used the fact that for some r independent of k. In fact, it is easy to verify that

Denoting

and combining (3.5) with (3.6), we have

Then

From the periodicity of , we know that there exists at which reaches its maximum and thus the left-hand side of (3.8) vanishes. Then we obtain

where

Therefore, we conclude that

Since and and are bounded, we get

where is a positive constant independent of k. Then we have

thus

or

where

Letting , we obtain

Now, we just need to show the estimate of . Multiplying (2.2) by u and integrating by parts over , by the periodicity of u, we have

which implies that

Let , by assumption (A2), we have

that is,

where C is a positive independent of λ. By Young’s inequality, we have

Combining with the above inequality, we have

which together with (3.9) implies (3.1), and thus the proof is complete. □

Corollary 1There exists a positive constant R independent ofεsuch that

whereis a ball centered at the origin with radiusRin.

Proof It follows from Lemma 2 that there exists a positive constant R independent of λ, ε such that

So, the degree is well defined on . From the homotopy invariance of the Leray-Schauder degree and the existence and uniqueness of the solution of , we can see that

The proof is completed. □

Lemma 3There exist constantsandsuch that for any, , admits no nontrivial solutionusatisfying

whereris a positive constant independent ofε.

Proof By contradiction, let u be a nontrivial fixed point of satisfying . For any given , multiplying (2.2) by and integrating over , we have

By the periodicity of u, the first term on the left-hand side is zero. The second term on the left-hand side can be rewritten as

The third term of the left-hand side of equation (3.11) can be rewritten as

Then from (3.10), we obtain

From assumption (A1), we can see that

Since , we have

By an approaching process, we choose , where is the eigenvector of the first eigenvalue in (A3), and then we obtain

Thus, there exists such that , then

From assumption (A2), we can see that

holds for any sufficiently small r and ε, which is a contradiction to assumption (A3). The proof is complete. □

Corollary 2There exists a small positive constantwhich is independent ofε, τsuch that

whereis a ball centered at the origin with radiusrin.

Proof Similar to Lemma 3, we can see that there exists a positive constant independent of ε such that

So, the degree is well defined on . By Lemma 3.3, we can easily see that admits no solution in . Then, by the homotopy invariance of the Leray-Schauder degree, we have

The proof is completed. □

Now, we show the proof of the main result of this paper.

Proof of Theorem 1

Using Corollaries 1 and 2, we have

where R and r are positive constants and . Problem (2.2)-(2.4) admits a nonnegative nontrivial solution with . Combining with the regularity results [9] and a similar argument as in [10], we can prove that the limit function of is a nonnegative nontrivial periodic solution of problem (1.1)-(1.3). □

The authors declare that they have no competing interests.

RH, JS and BW carried out the proof of the main part of this article, BW corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.

This work is partially supported by the National Science Foundation of China (11271100, 11126222), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2011006), the Natural Sciences Foundation of Heilongjiang Province (QC2011C020) and also by the 985 project of Harbin Institute of Technology.

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