Asymptotic Behavior of Solutions of a Periodic Diffusion Equation

Jiebao Sun,Dazhi Zhang,Boying Wu

doi:10.1155/2010/597569

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.

Highlights

Δum u a − bu, x, t ∈ Ω × R, 1.1 u x, t 0, x, t ∈ ∂Ω × R, 1.2 u x, 0 u0 x, x ∈ Ω, 1.3 where m > 1, Ω is a bounded domain in Rn with smooth boundary ∂Ω, u0 x is a nonnegative bounded smooth function, a a x, t and b b x, t are positive continuous functions and of T -periodic T > 0 with respect to t
The term Δum models a tendency to avoid crowding and the reaction term u a − bu models the contribution of the population supply due to births and deaths; see 1
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

Summary

Introduction

We consider the following periodic degenerate parabolic equation:. 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. 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. We first establish the existence of nontrivial nonnegative periodic solutions by monotone iterative method. By which we obtain asymptotic behavior of nontrivial nonnegative solutions of the problem 1.1 – 1.3.

Preliminaries

Existence of Periodic Solutions

Asymptotic Behavior