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# Periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions

- Raad Awad Hameed
^{1, 2}, - Boying Wu
^{1}and - Jiebao Sun
^{1}Email author

**2013**:34

https://doi.org/10.1186/1687-2770-2013-34

© Hameed et al.; licensee Springer. 2013

**Received:**26 November 2012**Accepted:**2 February 2013**Published:**21 February 2013

## Abstract

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.

## Keywords

- Periodic Solution
- Dirichlet Boundary Condition
- Neumann Boundary Condition
- Auxiliary Problem
- Nonnegative Solution

## 1 Introduction

where Ω is a bounded domain in ${\mathbb{R}}^{n}$ with smooth boundary *∂* Ω, $\frac{\partial}{\partial n}$ denotes the outward normal derivative on *∂* Ω, ${Q}_{T}=\mathrm{\Omega}\times (0,T)$, ${a}_{ij}$ 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 $u(x,t)$ 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 ${D}_{i}({a}_{ij}(x,t,u){D}_{j}u)$ models a tendency to avoid high density in the habitat, $m-\mathrm{\Phi}[u]$ 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.

*p*-Laplacian equation [7] with a typical form

where $m>1$. 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 [8], 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* $u\in {L}^{2}(0,T;{H}^{1}(\mathrm{\Omega}))\cap {C}_{T}(\overline{{Q}_{T}})$.

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.

## 2 Preliminaries

In the paper, we assume that

where
is the class of functions which are continuous in $\overline{\mathrm{\Omega}}\times \mathbb{R}$ and *T*-periodic with respect to *t*. Furthermore, ${a}_{ij}(\cdot ,\cdot ,u)$ is continuous with respect to *u*.

where *C* is a positive constant independent of *u*, ${\mathbb{R}}^{+}=[0,+\mathrm{\infty})$, ${L}_{+}^{2}(\mathrm{\Omega})=\{u\in {L}^{2}(\mathrm{\Omega})|u\ge 0,\text{a.e. in}\mathrm{\Omega}\}$.

where ${\lambda}_{1}$ is the first eigenvalue of the Laplacian equation on *ω* with zero boundary and ${\varphi}_{1}(x)$ 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.

**Definition 1**A function

*u*is said to be a weak solution of problem (1.1)-(1.3), if $u\in {L}^{2}(0,T;{H}^{1}(T))\cap {C}_{T}({\overline{Q}}_{T})$ and satisfies

for any $\phi \in {C}^{1}({\overline{Q}}_{T})$ with $\phi (x,0)=\phi (x,T)$.

where *ε* is a sufficiently small positive constant, $\tau \in [0,1]$ is a parameter and $f\in {C}_{T}(\overline{{Q}_{T}})$. Then we can define a map ${u}_{\epsilon}=G(\tau ,f)$ with $G:[0,1]\times {C}_{T}({\overline{Q}}_{T})\to {C}_{T}({\overline{Q}}_{T})$. Applying classical estimates (see [9]), we can see that ${\parallel {u}_{\epsilon}\parallel}_{{L}^{\mathrm{\infty}}({Q}_{T})}$ is bounded by ${\parallel f\parallel}_{{L}^{\mathrm{\infty}}({Q}_{T})}$, and ${u}_{\epsilon}$ is Hölder continuous in ${Q}_{T}$. Then, by the Arzela-Ascoli theorem, the map *G* is compact. So, the map *G* is a compact continuous map. Let $f({u}_{\epsilon})=(m-\mathrm{\Phi}[{u}_{\epsilon}]){u}_{\epsilon}^{+}$, where ${u}_{\epsilon}^{+}=max\{{u}_{\epsilon},0\}$, we can see that the nonnegative solution ${u}_{\epsilon}$ of problem (2.2)-(2.4) is also a nonnegative fixed point of the map ${u}_{\epsilon}=G(1,(m-\mathrm{\Phi}[{u}_{\epsilon}]){u}_{\epsilon}^{+})$. So, we will study the existence of nonnegative fixed points of the map $u=G(1,(m-\mathrm{\Phi}[{u}_{\epsilon}]){u}_{\epsilon}^{+})$ 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 ${u}_{\epsilon}$.

## 3 The proof of the main result

First, by the same method as in [4], we can obtain the nonnegativity of the solutions of problem (2.2)-(2.4).

**Lemma 1**

*If a nontrivial function*${u}_{\epsilon}\in {C}_{T}({\overline{Q}}_{T})$

*solves*${u}_{\epsilon}=G(1,(m-\mathrm{\Phi}[{u}_{\epsilon}]){u}_{\epsilon}^{+})$,

*then*

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 ${\parallel \cdot \parallel}_{p}$ ($1\le p\le \mathrm{\infty}$) the ${L}^{p}(\mathrm{\Omega})$ norm.

**Lemma 2**

*For*$\lambda \in [0,1]$,

*let*$u(x,t)$

*be a nonnegative periodic solution which solves*${u}_{\epsilon}=G(1,\lambda (m-\mathrm{\Phi}[{u}_{\epsilon}]){u}_{\epsilon}^{+})$,

*then there exists a constant*

*K*

*independent of*

*λ*,

*ε*

*such that*

*where* $u(t)=u(\cdot ,t)$.

*Proof*Multiplying (2.2) by ${u}^{m+1}$ ($m\ge 0$) and integrating over Ω, we have

*u*and

*m*. Assume that ${\parallel u(t)\parallel}_{\mathrm{\infty}}\ne 0$ and set

*C*a positive constant independent of

*k*and

*m*, which may take different values. From (3.2), we obtain

*r*independent of

*k*. In fact, it is easy to verify that

*k*. Then we have

*u*and integrating by parts over ${Q}_{T}$, by the periodicity of

*u*, we have

*C*is a positive independent of

*λ*. By Young’s inequality, we have

which together with (3.9) implies (3.1), and thus the proof is complete. □

**Corollary 1**

*There exists a positive constant R independent of*

*ε*

*such that*

*where* ${B}_{R}$ *is a ball centered at the origin with radius* *R* *in* ${L}^{\mathrm{\infty}}({Q}_{T})$.

*Proof*It follows from Lemma 2 that there exists a positive constant

*R*independent of

*λ*,

*ε*such that

The proof is completed. □

**Lemma 3**

*There exist constants*${r}_{0}>0$

*and*$\epsilon >0$

*such that for any*$r<{r}_{0}$, $\epsilon <{\epsilon}_{0}$, $u=G(\tau ,(m-\mathrm{\Phi}[u]){u}^{+}+(1-\tau ))$

*admits no nontrivial solution*

*u*

*satisfying*

*where* *r* *is a positive constant independent of* *ε*.

*Proof*By contradiction, let

*u*be a nontrivial fixed point of $u=G(\tau ,(m-\mathrm{\Phi}[u]){u}^{+}+1-\tau )$ satisfying $0<{\parallel u\parallel}_{{L}^{\mathrm{\infty}}({Q}_{T})}\le r$. For any given $\varphi (x)\in {C}_{0}^{\mathrm{\infty}}({B}_{\delta}({x}_{0}))$, multiplying (2.2) by $\frac{{\varphi}^{2}}{u}$ and integrating over ${Q}_{T}^{\ast}={B}_{\delta}({x}_{0})\times (0,T)$, we have

*u*, the first term on the left-hand side is zero. The second term on the left-hand side can be rewritten as

holds for any sufficiently small *r* and *ε*, which is a contradiction to assumption (A3). The proof is complete. □

**Corollary 2**

*There exists a small positive constant*$r<R$

*which is independent of*

*ε*,

*τ*

*such that*

*where* ${B}_{r}$ *is a ball centered at the origin with radius* *r* *in* ${L}^{\mathrm{\infty}}({Q}_{T})$.

*Proof*Similar to Lemma 3, we can see that there exists a positive constant $0<r<R$ independent of

*ε*such that

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 $R>r$. Problem (2.2)-(2.4) admits a nonnegative nontrivial solution ${u}_{\epsilon}$ with $r\le {\parallel {u}_{\epsilon}\parallel}_{\mathrm{\infty}}\le R$. Combining with the regularity results [9] and a similar argument as in [10], we can prove that the limit function of ${u}_{\epsilon}$ is a nonnegative nontrivial periodic solution of problem (1.1)-(1.3). □

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

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