Periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions
© Hameed et al.; licensee Springer. 2013
Received: 26 November 2012
Accepted: 2 February 2013
Published: 21 February 2013
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.
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.
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 , 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 1 If 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
where is the class of functions which are continuous in and T-periodic with respect to t. Furthermore, is continuous with respect to u.
where C is a positive constant independent of u, , .
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.
for any with .
where ε is a sufficiently small positive constant, is a parameter and . Then we can define a map with . Applying classical estimates (see ), 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 .
3 The proof of the main result
First, by the same method as in , we can obtain the nonnegativity of the solutions of problem (2.2)-(2.4).
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.
which together with (3.9) implies (3.1), and thus the proof is complete. □
where is a ball centered at the origin with radius R in .
The proof is completed. □
where r is a positive constant independent of ε.
holds for any sufficiently small r and ε, which is a contradiction to assumption (A3). The proof is complete. □
where is a ball centered at the origin with radius r in .
The proof is completed. □
Now, we show the proof of the main result of this paper.
Proof of Theorem 1
where R and r are positive constants and . Problem (2.2)-(2.4) admits a nonnegative nontrivial solution with . Combining with the regularity results  and a similar argument as in , we can prove that the limit function of is a nonnegative nontrivial periodic solution of problem (1.1)-(1.3). □
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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