PERIODIC SOLUTION FOR A CLASS OF DOUBLY DEGENERATE PARABOLIC EQUATION WITH NEUMANN PROBLEM

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


INTRODUCTION
The goal of the present text is to consider the boundary conditions in equations (1.1) to (1.3) for periodic doubly degenerate parabolic equation with Neumann boundary. models the tendency to avoid high density in the habitat. As population growing is controlled by birth, death, emigrant, and immigration, assumption of m, [ ] hould be made to describe the ways in which a given population grows and shrinks over time.
Recently, periodic problems with nonlocal terms have been investigated intensively by number of researchers [1][2][3][4][5]. A typical model was submitted by Allegretto and Nistri in which they with Dirichlet boundary conditions. Also, according to the actual needs, many authors diverts attention to nonlinear diffusion equations with nonlocal terms such as the porous equation [6,7] with typical form: And a class double degenerate parabolic equation [8] with the typical from shown in equation (1.5). By comparing the doubly degenerate parabolic equation with Dirichlet boundary equation, the Neumann boundary condition causes an additional difficulty in establishing a priori estimate. On the other hand, different form the case of Dirichlet boundary condition, the auxiliary problem in equations (1.1) to (1.3) is considered for using the theory of Leray-Schauder degree. We have proved that the problem in equations (1.1) to (1.3) admits a non-trivial nonnegative periodic solution as shown in the following theorem.
The rest of this article is organized as follows: In Section 2, we present some necessary preliminaries including the auxiliary problem. in section 3, we establish the necessary priori estimations of the solution of the auxiliary problem. Then the proof of the main result of this article is shown in the last section.

Preliminaries
In this paper, we assume that:  R are the boundary conditions functional satisfying the condition: 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 the problem (1. In order to use the theory of Leray-Schauder degree, we introduce a map by considering the following auxiliary problem ) , Then we can define a map u Gf   with : ( ) ( ).
TT TT G C Q C Q  by applying classical estimated (see [9]), we can know that Then by the Arzela-Ascoli theorem, the map G is compact. So the map is a compact continuous map. Let . So we will study the existence of the nonnegative fixed points of the map 3. Proof of the main results :First, by the same way as in [5], we can get the non-negativity of the solution of problem (2.2)-(2.4) .

Lemma 1 If a nontrivial function
In the following, by the Moser iterative technique, we will show the priori estimate for the upper bound of nonnegative periodic solution of problem (2.5)-(2.7). Here and below we denote by . (1 ) And hence:  We can using the Gagliardo-Nirenberg inequality, we have By inequalities (3.4)-(3.5) and the fact that we obtain the following differential inequality: Here we have used the fact that '1 k p l r    for some r independent of k. in fact, it is easy to verify that lim .
Where ( 1,2) i Ci  are constant independent of u. Taking the periodicity of u into account, we infer from (3.14) that B is a ball centered at the origin with radius R in ( ).

T LQ 
Proof it follows from Lemma 2 that there exists appositive constant R independent of ε, such that

 
So the degree is will defined on R B . from the homotopy invariance of the Leray-schauder degree and the existence and uniqueness of the solution of G(1,0) , we can see that The proof is completed. .

Corollary 2
There exists a small positive constant r which is independent of  and satisfies rR  such that