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 LeraySchauder degree, we obtain the existence of a nontrivial nonnegative time periodic solution.
1 Introduction
The aim of this work is to consider the following periodic problem for a quasilinear parabolic equation:
where Ω is a bounded domain in
In last decades, linear parabolic equations with nonlocal terms have been investigated by numerous researchers [14]. A typical model was submitted by Allegretto and Nistri [1] and they proposed the following equation:
with the Dirichlet boundary conditions. Also, according to the actual needs, many authors divert attention to nonlinear diffusion equations with nonlocal terms such as the porous equation [5,6] with a typical form
and the pLaplacian equation [7] with a typical form
Equation (1.4) is degenerate if
where
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 LeraySchauder degree. We prove that this problem (1.1)(1.3) admits a nontrivial nonnegative periodic solution, that is, the following theorem.
Theorem 1If 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.
2 Preliminaries
In the paper, we assume that
(A1)
where is the class of functions which are continuous in
(A2)
where C is a positive constant independent of u,
(A3)
where
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
for any
In order to use the theory of LeraySchauder degree, we introduce a map by considering the following auxiliary problem:
where ε is a sufficiently small positive constant,
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 1If a nontrivial function
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
Lemma 2For
where
Proof Multiplying (2.2) by
and hence
where
then
By using the GagliardoNirenberg inequality, we have
with
By inequalities (3.3), (3.4) and the fact that
Let
we have
By Young’s inequality,
where
then we obtain
Here we have used the fact that
Denoting
and combining (3.5) with (3.6), we have
Then
From the periodicity of
where
Therefore, we conclude that
Since
where
thus
or
where
Letting
Now, we just need to show the estimate of
which implies that
Let
that is,
where C is a positive independent of λ. By Young’s inequality, we have
Combining with the above inequality, we have
which together with (3.9) implies (3.1), and thus the proof is complete. □
Corollary 1There exists a positive constant R independent ofεsuch that
where
Proof It follows from Lemma 2 that there exists a positive constant R independent of λ, ε such that
So, the degree is well defined on
The proof is completed. □
Lemma 3There exist constants
whereris a positive constant independent ofε.
Proof By contradiction, let u be a nontrivial fixed point of
By the periodicity of u, the first term on the lefthand side is zero. The second term on the lefthand side can be rewritten as
The third term of the lefthand side of equation (3.11) can be rewritten as
Then from (3.10), we obtain
From assumption (A1), we can see that
Since
By an approaching process, we choose
Thus, there exists
From assumption (A2), we can see that
holds for any sufficiently small r and ε, which is a contradiction to assumption (A3). The proof is complete. □
Corollary 2There exists a small positive constant
where
Proof Similar to Lemma 3, we can see that there exists a positive constant
So, the degree is well defined on
The proof is completed. □
Now, we show the proof of the main result of this paper.
Proof of Theorem 1
Using Corollaries 1 and 2, we have
where R and r are positive constants and
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
RH, JS and BW carried out the proof of the main part of this article, BW corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.
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.
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