- Research Article
- Open Access
Asymptotic Behavior of Solutions of a Periodic Diffusion Equation
© Jiebao Sun et al. 2010
- Received: 25 September 2009
- Accepted: 24 December 2009
- Published: 18 January 2010
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.
- Periodic Solution
- Weak Solution
- Global Existence
- Diffusion Equation
- Initial Boundary
In this paper, we consider the following periodic degenerate parabolic equation:
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 . 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).
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.
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.
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 .
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.
Local existence can be proved as . Global existence and nonnegativity follow from Lemma 2.3 by standard arguments.
Lemma 2.5 (regularity ).
for every pair of points .
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 ).
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.
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.
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 .
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.
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 .
Together with (4.17), we complete the proof of this lemma.
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).
Similar to the proof of Theorem in , 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.
Thus we complete the proof.
The authors express their thanks to the referees for their very helpful suggestions to improve some results in this paper. This work is supported by NSFH (A200909), NSFC (10801061), and Project (HIT. NSRIF. 2009049). It was also supported by the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology.
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