Periodic solutions of a quasilinear parabolic equation with nonlinear convection terms
© Li and Hui; licensee Springer 2012
Received: 26 November 2011
Accepted: 14 November 2012
Published: 28 November 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.
where Ω is a bounded domain in with a smooth boundary ∂ Ω, , and we assume 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.
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.
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.
when the measure of Ω is small enough.
where , are positive constants independent of , p.
If , from (2.7) we can get (2.9) directly.
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.
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.
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. □
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).
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