Asymptotic Behavior of Solutions of a Periodic Diffusion Equation

Abstract

We consider a degenerate parabolic equation with logistic periodic sources. First, we establish the existence of nontrivial nonnegative periodic solutions by monotonicity method. Then by using Moser iterative technique and the method of contradiction, we establish the boundedness estimate of nonnegative periodic solutions, by which we show that the attraction of nontrivial nonnegative periodic solutions, that is, all non-trivial nonnegative solutions of the initial boundary value problem, will lie between a minimal and a maximal nonnegative nontrivial periodic solutions, as time tends to infinity.

1. Introduction

In this paper, we consider the following periodic degenerate parabolic equation:

(1.1)

(1.2)

(1.3)

where ,   is a bounded domain in with smooth boundary ,   is a nonnegative bounded smooth function, and are positive continuous functions and of -periodic with respect to .

The problem (1.1)–(1.3) describes the evolution of the population density of a species living in a habitat and can be proposed for many problems in mathematical biology and fisheries management. The term models a tendency to avoid crowding and the reaction term models the contribution of the population supply due to births and deaths; see [1]. The homogeneous Dirichlet boundary conditions model the inhospitality of the boundary. The time dependence of the coefficients reflects the fact that the time periodic variations of the habitat are taken into account. Reaction diffusion equations with such reaction term can be regarded as generalization of Fisher or Kolomogorv-Petrovsky-Piscunov equations which are used to model the growth of population (see [2, 3]). Especially, when , the (1.1) is the classical Logistic equation and some related problems have attracted much attention of researchers (see [4–6], etc.).

In the recent years, there are a lot of work dedicated to the existence, uniqueness, regularity, and some other qualitative properties, of weak solutions of this kind of degenerate parabolic equations (see [7–9], etc.). But to our knowledge, there is few work that has been accomplished in the literature for periodic degeneracy parabolic equation, and most of the known results so far only concerned with the existence of periodic solutions but not consider the attraction (see [10, 11], etc.). So our work is not a simple extension to the previous work.

The purpose of this paper is to investigate the asymptotic behavior of nontrivial nonnegative solutions of the initial boundary value problem (1.1)–(1.3). Since the equation has periodic sources, it is of no meaning to consider the steady state. So we have to seek some new approaches. Our idea is to consider all nonnegative periodic solutions. We first establish the existence of nontrivial nonnegative periodic solutions by monotone iterative method. Then we establish the a priori upper bound and a priori lower bound according to the maximum norm for all nontrivial nonnegative periodic solutions. By which we obtain asymptotic behavior of nontrivial nonnegative solutions of the problem (1.1)–(1.3). That is all nontrivial nonnegative solutions will lie between a minimal and a maximal nonnegative nontrivial periodic solutions, as time tends to infinity.

The paper is organized as follows. In Section 2, we introduce some necessary preliminaries. In Section 3, we establish the existence of nontrivial nonnegative periodic solutions by monotonicity method. In Section 4, we show the asymptotic behavior of nontrivial nonnegative solutions of (1.1)–(1.3).

2. Preliminaries

In this section, we present the definitions of weak solutions and some useful principles.

Since (1.1) is degenerate at points where , problem (1.1)–(1.3) might not have classical solutions in general. Therefore, we focus our main efforts on the discussion of weak solutions in the sense of the following.

Definition 2.1.

A nonnegative function is called to be a weak solution of problem (1.1)–(1.3) in , if and satisfies

(2.1)

for any test functions with , where denotes the set of functions which are continuous in and of -periodic with respect to .

A supersolution (resp., a subsolution ) is defined in the same way except that the "" in (2.1) is replaced by "" ("") and is taken to be nonnegative.

Definition 2.2.

A function is called to be a periodic solution of problem (1.1)-(1.2) if it is a solution such that . A function is called to be a periodic subsolution if it is a subsolution such that in . A function is called to be a periodic supersolution if it is a supersolution such that in . A pair of periodic supersolution and subsolution is called to be ordered if in .

Several properties of solutions of problem (1.1)–(1.3) are needed in this paper. We first show the comparison principle.

Lemma 2.3 (comparison).

Assume , if is a subsolution of (1.1)–(1.3) corresponding to the initial datum , and is a supersolution of (1.1)–(1.3) corresponding to the initial datum , then .

Proof.

Without loss of generality, we might assume that are bounded. From Definition 2.1, we have

(2.2)

with nonnegative test function and . Subtracting the above inequalities, we get

(2.3)

where

(2.4)

Since , and are bounded on , it follows from that is a bounded nonnegative function. Thus, by choosing appropriate test function exactly as [12, Pages 118–123], we obtain

(2.5)

where and is a bounded constant. Since , combining with the Gronwall's lemma, we see that a.e. in for any . The proof is completed.

Lemma 2.4 (global existence).

For any nonnegative bounded initial value , problem (1.1)–(1.3) admits a global nonnegative solution.

Proof.

Local existence can be proved as [13]. Global existence and nonnegativity follow from Lemma 2.3 by standard arguments.

Lemma 2.5 (regularity [7]).

Let be a weak solution of problem (1.1)–(1.3), then there exist positive constants and , such that

(2.6)

for every pair of points .

3. Existence of Periodic Solutions

In this section, we show the existence of nontrivial nonnegative periodic solutions of the problem (1.1)-(1.2) by monotonicity method. First, we introduce the following remark.

Remark 3.1 (see [14]).

According to Lemmas 2.3–2.5, the semiflow associated with the solution of problem (1.1)–(1.3), namely, the map

(3.1)

has the following properties:

(i) is well defined for any (Lemma 2.3);

(ii) is order preserving (Lemma 2.4);

(iii) is compact. In fact, the family    is uniformly bounded in by Lemma 2.3. Then by Lemma 2.5, the set consists of equicontinuous functions, thus the conclusion follows from Ascoli-Arzelà's theorem.

Theorem 3.2.

If problem (1.1)-(1.2) admits a pair of ordered nontrivial nonnegative periodic subsolution and periodic supersolution , then problem (1.1)-(1.2) admits a nontrivial nonnegative periodic solutions.

Proof.

From Remark 3.1, we just need to construct a pair of ordered periodic subsolution and periodic supersolution. The existence of nontrivial nonnegative periodic solutions of problem (1.1)-(1.2) will come from the similar iteration procedure as that in [15].

Let be the first eigenvalue and its corresponding eigenfunction to the Laplacian operator on the domain the first eigenvalue and its corresponding eigenfunction to the Laplacian operator on some domain , with respect to homogeneous Dirichlet data, respectively. It is clear that for all . Denote

(3.2)

and define

(3.3)

where

(3.4)

Clearly, and are the -periodic subsolution and supersolution of (1.1) subject to the condition (1.2), respectively. Further, we may assume , else we may change and then appropriately. Thus we complete the proof.

4. Asymptotic Behavior

In this section, we show the asymptotic behavior of nontrivial nonnegative solutions of the initial boundary value problem. First, we employ Moser's technique to obtain the upper bound of norm for a nonnegative periodic solution .

Lemma 4.1.

Let be a nontrivial nonnegative periodic solution of (1.1)-(1.2), then there exists a positive constant which is independent of , such that

(4.1)

where .

Proof.

Let be a nontrivial nonnegative periodic solution of (1.1)-(1.2), multiply the (1.1) by and integrate the resulting relation over , we have

(4.2)

where . Hence

(4.3)

where denotes various positive constants independent of and . Set

(4.4)

from (4.3) we have

(4.5)

Here we appeal to the Gagliardo-Nirenberg inequality

(4.6)

with

(4.7)

where denotes a positive constant independent of and . From (4.5), (4.6), and the fact that , we have

(4.8)

with . Taking the periodicity of into account, we can obtain

(4.9)

where . From the boundedness of and , we can see that

(4.10)

where is a positive constant independent of . Noticing that implies

(4.11)

with , we get

(4.12)

where . That is

(4.13)

and thus

(4.14)

or

(4.15)

where

(4.16)

Letting , we get

(4.17)

Now we estimate . Set in (4.3), we get

(4.18)

By Hölder's inequality and Sobolev's theorem, we have

(4.19)

From (4.18), (4.19) we can obtain

(4.20)

By Young's inequality, we get

(4.21)

where denotes different positive constants independent of . By (4.21) and the periodicity of , we have

(4.22)

Together with (4.17), we complete the proof of this lemma.

Lemma 4.2.

There exists a constant with , such that no nontrivial nonnegative periodic solutions of problem (1.1)-(1.2) satisfies

(4.23)

Proof.

To arrive at a contradiction, we assume that problem (1.1)-(1.2) admits a nontrivial nonnegative periodic solution satisfying . For any given , we can choose as a test function. Multiplying (1.1) by and integrating over , we obtain

(4.24)

By the periodicity of , the first term of the left-hand side in (4.24) satisfies

(4.25)

The second term of the left-hand side in (4.24) can be rewritten as

(4.26)

Thus

(4.27)

Combining (4.24) with (4.25) and (4.27), we obtain

(4.28)

By an approximating process, we can choose with is the first eigenvalue and is its corresponding eigenfunction to the eigenvalue problem

(4.29)

Then and is strictly positive in . From (4.28) we have

(4.30)

Thus there exists such that , then

(4.31)

Obviously, we can choose suitable small , such that for any , the above inequality does not hold. It is a contradiction. The proof is completed.

In the following, we will make use of the a priori boundedness of all nontrivial nonnegative periodic solutions to show the asymptotic behavior of nontrivial nonnegative solutions of the initial boundary value problem (1.1)–(1.3).

Theorem 4.3.

Problem (1.1)-(1.2) admits a minimal and a maximal nonnegative nontrivial periodic solutions and . Moreover, if is the solution of the initial boundary value problem (1.1)–(1.3) with initial value , then for any , there exists depending on and , such that

(4.32)

Proof.

First, we show the existence of the maximal periodic solution . Define a Poincaré map

(4.33)

where is the solution of (1.1)–(1.3) with initial value . By Remark 3.1, the map is well defined. Let be the solution of (1.1)–(1.3) with initial value , where and is a positive constant satisfying

(4.34)

where are chosen as those in Theorem 3.2. It is observed that , , and by the comparison principle. By a rather standard argument, we conclude that there exist a function and a subsequence of , denoted by itself for simplicity, such that

(4.35)

Similar to the proof of Theorem in [4], we can prove that , which is the even extension of the solution of the initial boundary value problem (1.1)–(1.3) with initial value , is a periodic solution of the problem (1.1)-(1.2). Moreover, Lemma 4.1 shows that any nonnegative periodic solution of (1.1)-(1.2) must satisfy for . Therefore, if we take is larger than by the comparison principle we have and thus , which means that is the maximal periodic solution of problem (1.1)-(1.2). The existence of the minimal periodic solution can be obtained with the same method.

Let be the solution of the initial boundary value problem (1.1)–(1.3) with any given nonnegative initial value , and let be the solution of (1.1)–(1.3) with initial value , where is a positive constant satisfying

(4.36)

then for any , we have

(4.37)

A similar argument as [4] shows that , and is a nontrivial nonnegative periodic solution of (1.1)-(1.2). Therefore, there exists such that

(4.38)

for any and . Provided that the periods of and are taken into account, for any , there exists depending on and such that

(4.39)

By the similar way and Lemma 4.2, we can obtain

(4.40)

Thus we complete the proof.