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Periodic solutions of a quasilinear parabolic equation with nonlinear convection terms
Advances in Difference Equations volume 2012, Article number: 206 (2012)
In this paper, we study a periodic quasilinear parabolic equation with nonlinear convection terms and weakly nonlinear sources. Based on the theory of the Leray-Schauder fixed point theorem, we establish the existence of periodic solutions when the domain of the solution is sufficiently small.
In this paper, we consider the following periodic quasilinear parabolic equation with nonlinear convection terms and weakly nonlinear sources:
where Ω is a bounded domain in with a smooth boundary ∂ Ω, , and we assume that
(A1) and there exist two constants such that
(A2) is Hölder continuous in , periodic in t with a period T and satisfies with constants and .
(A3) , for , where denotes the set of functions which are continuous in and ω-periodic with respect to t.
The existence of periodic solutions for parabolic equations has been considered by several authors; see [1–12] and the references therein. As a work related to this paper, we refer to Nakao , in which the author considered the following parabolic equation:
with Dirichlet boundary value conditions, where B, h are periodic in t with a period , satisfies except for and is fulfilled by if . Under the assumption that , Nakao established the existence of periodic solutions by the Leray-Schauder fixed point theorem. In , Zhou et al. considered the quasilinear parabolic equation with nonlocal terms. Based on the theory of Leray-Schauder’s degree, the authors established the existence of nontrivial periodic solutions. In this paper, we consider the quasilinear parabolic equation (1.1) with weakly nonlinear sources and nonlinear convection terms. The convection term describes an effect of convection with a velocity field . Under a restrictive condition that the domain is sufficiently small, we establish the existence of periodic solutions of the problem (1.1)-(1.3).
This paper is organized as follows. The definition of the generalized solution and a useful a priori estimate are presented in Section 2. Our main results will be given in Section 3.
Our main efforts will focus on the discussion of generalized solutions since the regularity follows from a quite standard approach. Hence, we give the following definition of generalized solutions.
Definition 1 A function u is said to be a generalized solution of the problem (1.1)-(1.3) if and
for any with and , where and , .
For convenience, we let and denote and norms, respectively. First, we establish the following a priori estimate which plays an important role in the proof of the main results of this paper.
Lemma 1 Let u be a solution of
with , then there exists a positive constant R independent of σ such that
when the measure of Ω is small enough.
Proof Suppose u is a solution of the problem (2.2)-(2.4). Multiplying equation (2.2) by () and integrating the resulting relation over Ω, noticing that
where n is the outer normal to ∂ Ω, we have
The second term of the left-hand side in the above integral equality can be written as
Hence, from (2.6), we have
where , are positive constants independent of , p.
If , by Hölder’s inequality and Young’s inequality, we have
Combined with (2.7), it yields
If , from (2.7) we can get (2.9) directly.
then . From (2.9), we have
By the Gagliardo-Nirenberg inequality, we have
Noticing , by (2.10) we obtain
Set , then
Now, we estimate . By Young’s inequality,
where , , with
It is easy to see that . Denote
From (2.12), (2.13), we have
The periodicity of implies that there exists such that takes its maximum and the left-hand side of (2.14) vanishes. Then we have
Since (), it follows that
Noticing that and are bounded, we have
where is a positive constant independent of k. That is,
where , then
Letting , we obtain
In order to estimate , we set . From (2.9), we obtain
According to the Poincaré inequality, we have
where is a positive constant which depends only on N and the measure of Ω and becomes very large when becomes small. Then
So, when is sufficiently small, we have . Then by Young’s inequality, we obtain
where C is a constant independent of u. By the periodicity of u, we have
where R is a positive constant independent of σ. Combining the above inequality with (2.15), we obtain (2.5). The proof is completed. □
3 The main results
Our main result is the following theorem.
Theorem 1 If (A1), (A2) and (A3) hold, then the problem (1.1)-(1.3) admits at least one periodic solution u.
Proof First, we define a map by considering the following problem:
where is a given function in . It follows from a standard argument similar to  that the problem (3.1)-(3.3) admits a unique solution. So, we can define a map by and the map is compact and continuous. In fact, by the method in , we can infer that is bounded if and u, for some . Then (by the Arzela-Ascoli theorem) the compactness of the map T comes from and the Hölder continuity of u. The continuity of the map T comes from the Hölder continuity of ∇u.
Let , by (A2)-(A3) and the above arguments, we see that is a complete continuous map for . By Lemma 1, we can see that any fixed point u of the map satisfies
where C is a positive constant independent of σ. Then, by the Leray-Schauder fixed point theorem , we conclude that the problem (1.1)-(1.3) admits a periodic solution u. The proof is complete. □
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The authors are grateful to the associate editor and the referees for their helpful suggestions to improve some results in this paper. This work is supported by the National Natural Science Foundation of China (71173060, 70773028, 71031003) and the Fundamental Research for the Central Universities (Grant No. HIT. HSS. 201201).
The authors declare that they have no competing interests.
SL and XH carried out the proof of the main part of this article, XH corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.