Periodic solutions of a porous medium equation

In this paper, we study with a periodic porous medium equation with nonlinear convection terms and weakly nonlinear sources under Dirichlet boundary conditions. Based on the theory of Leray-Shauder fixed point theorem, we establish the existence of periodic solutions.

In recent years, periodic problems for degenerate parabolic equations have been the subject of extensive study, see [2,5,7,9,11,12,13] and references therein.Among the earliest works of this aspect, we refer to Nakao [9], in which one can find the related result for the special case of the equation (1.1), that is with Dirichlet boundary value conditions, where B, h are periodic in t with period ω > 0, β(u) satisfies β ′ (u) > 0 except for u = 0 and β(u) is fulfilled by Under the assumption that B(x, t, u)u ≤ b 0 |u|, Nakao established the existence of periodic solutions by Leray-Shauder fixed point theorem.In [12], Wang et al. considered the following porous medium equation with weakly nonlinear sources By Moser's technique and the Leray-Schauder fixed point theorem [6, Th2.1, pp.140], the authors established the existence of periodic solutions when the assumption (A2) holds .The work of this paper is an extension of [9,12], that is, we consider the porous medium equation (1.1) with weakly nonlinear sources and nonlinear convection terms.The convection term b(u) • ∇u describes an effect of convection with a velocity field b(u).Our aim is to establish the existence of periodic solutions of the equation (1.1) under Dirichlet boundary value conditions.
This paper is organized as follows: In Section 2, we state some necessary preliminaries including the definition of the generalized solution, some useful lemmas and the statement of the main results.In Section 3, we show the proof of the main results of this paper.

Preliminaries
Due to the degeneracy of the equation considered, the problem (1.1)-(1.3)admits no classical solutions in general, so we consider generalized solutions in the following sense.Definition 2.1.A function u is said to be a generalized solution of the problem (1.1)- and for any ϕ ∈ C 1 (Q ω ) with ϕ(x, 0) = ϕ(x, ω) and ϕ| ∂Ω×(0,ω) = 0, we have where For convenience, we let • p and • m,p denote L p (Ω) and W m,p (Ω) norms, respectively.In the following, we introduce some useful lemmas which play an important role in the proof of the main results of this paper. ) , where C is a constant independent of q, r, β and θ.
[10] Let y(t) ∈ C 1 (R 1 ) be a nonnegative ω periodic function satisfying the differential inequality with some α, A > 0, B ≥ 0 and C ≥ 0, then Our results will be proved by means of parabolic generalization, that is we consider the following regularized problem where ε is some positive constant.We will apply the Leray-Schauder fixed point theorem to establish the existence of the solution u ε of the problem (2.3)-(2.5).The desired solution of the problem (1.1)-( 1.3) will be obtained as a limit point of u ε .
Our main results is the following theorem.3) admits at least one periodic solution u.

Proof of the Main Results
First, we establish the following a priori estimate.
Lemma 3.1.Let u ε be a solution of ) where u ε (t) = u ε (•, t) and R is a positive constant independent of ε and σ.
where n is the outer normal to ∂Ω, we have Notice that the second term on the left hand can be written by and , from (3.5) we have Then we have where C 1 , C 2 are positive constants independent of u ε (t), p. First, we consider the case of 1 ≤ α < m.If N > 2, by Hölder's inequality, we have . By Sobolev's imbeding theorem in [1,Th5.4,pp.114] and Young's inequality, we have (3.9) EJQTDE, 2011 No. 42, p. 4 Combining (3.6) with (3.9), we have where C 1 , C 2 are positive constants independent of u, p.
By Gagliardo-Nirenberg inequality, we have (3.13) Set p + 2 = k i in (3.12), by (3.13) we have We set without loss of generality that B i > 1, which implies . It is easy to verify that {α i } is bounded (see [9]), and The proof is completed.Now, we show the proof of the main results.Proof of Theorem 2.1 First, we introduce a map by considering the following problem where g ∈ C ω (Q ω ).With a similar method of [9], we conclude that for any g Then we can infer that the map u ε = T g is compact and continuous.
In fact, by [4] and the periodicity of u ε , we have for every pair of points (x 1 , t 1 ), (x 2 , t 2 ) ∈ Q ω , where the constants γ, β ∈ (0, 1) are independent of ε.By Ascoli-Arezela theorem, we can see that T maps any bounded set of C ω (Q ω ) into a compact set of C ω (Q ω ).Suppose that g k → g as k → ∞ and denote u k = T g k , then there exist a subsequence of u k and a function u ε ∈ C ω (Q ω ) such that u k (x, t) → u ε (x, t), uniformly in Q ω .
Noticing that So we can prove that u ε = T g is compact and continuous by using the argument similar to [13].