Periodic Solutions of p-Laplacian Systems with a Nonlinear Convection Term

In this work, we study the existence of periodic solutions for the evolution of p-Laplacian system and we show that these periodic solutions belong to L ∞ (ω, W 1,∞ (Ω)) and give a bound of ∇u i (t) ∞ under certain geometric conditions on ∂Ω.

The system of form (S) is a class of degenerate parabolic systems and appears in the theory of non-Newtonian fluids perturbed by nonlinear terms and forced by rather irregular period in time excitations, see [1,5] .Periodic parabolic single equations have been the subject of numbers of extentive works see [3,5,11,12,14,15,16].In particular, Lui [11], has considered the following single equation: with Dirichlet boundary condition and has obtained the existence and gradient estimates of a periodic solution.The basic tools that have been used are a priori estimates, Leray-Shauder fixed point theorem, topological degree theory and Moser method.We first consider the parabolic regularisation of (S) and we employ Moser's technique as in [4] and we make some devices as in [7,13] to obtain the existence of periodic solutions and we derive estimates of ∇u i (t).

Main results
For convenience, hereafter let E = C ω (Q), the set of all functions in C(Ω×R) which are periodic in t with period w. .q and .s,q denote L q = L q (Ω) and W s,q = W s,q (Ω) norms, respectively, 1 ≤ q ≤ ∞.Since , Ω ⊂ R N is a bounded convex domain, we take the equivalent norm in W 1,q 0 (Ω) to be f or any u ∈ W 1,q 0 (Ω).
In the sequel, the same symbol c will be used to indicate some positive constants, possibly different from each other, appearing in the various hypotheses and computations and depending only on data.When we need to fix the precise value of one constant, we shall use a notation like M i , i = 1, 2....., instead.
We make the following assumptions: We make the following definition of solutions.Definition 2.1 A function u = (u 1 , u 2 ) is called a solution of system (S) if Let (H1) to (H4) be satisfied.Then there exists a solution (u 1 , u 2 ) of problem (S) such that for i = 1, 2, we have Theorem 1.2.Let (H1) to (H4) be satisfied.Then there exists a solution (u 1 , u 2 ) of problem (S) such that for i = 1, 2, we have For the proof of the theorems we use the following elementary lemmas.Lemma 2.1.
where C is a constant independent of q, r, β, and θ.
In this section we derive a priori estimates of solutions (u 1 , u 2 ) of problem (S).We first replace the p-Laplacian by the regularized one ∇u).
To prove theorem 1.1, we consider the following.By theorem 1 in [3] , we can choose u 0 iε such that : and we construct two sequences of functions (u n 1ε ) and (u n 2ε ), such that : u n 2ε = 0 on S, (1.5) It is clear that for each n = 1, 2, 3, ....., the above systems consist of two nondegenerated and uncoupled initial boundary-value problems.By theorem 1, [3] for fixed n and ε, the problem (1.1)-(1.3)and (1.4)-(1.6)has a solution (u n 1ε , u n 2ε ).We need lemma 2.3 and lemma 2.4 below to complete the proof of theorem 1.1.
Lemma 2.3.There exists a positive constant M i independent of ε and n, such that Proof.For n = 0, (1.7) is proved in [11], so suppose (1.7) for (n − 1).Multiplying (1.1) by |u n 1ε | k u n 1ε , k integer, and integrating over Ω to obtain : We note that ) , by Hölder's inequality and Lemma 2.1, we have where we set q = (k + p 1 )N/(N − p 1 ), . By lemma 2.1, we have Therefore, (1.17) Let λ l = sup t u n 1ε k l , by lemma 2.2, we obtain By [7], we have {λ l } bounded and we set without loss of generality that R > 1, which implies sup The same holds also for u n 2ε .Lemma 2.4.u n iε satisfies the following: By the periodicity, Hölder's inequality, Poincare's inequality and the L ∞ norm bounded of u n iε , we have in which r * = p 1 /(p 1 − 1 − α 1 ).Thus, we have By hypothesis and Young's inequality, we get By integration by parts, we have where we note that |∇v| = ∂v ∂n on ∂Ω and H ) is the mean curvature on ∂Ω.Using that Ω is convex so ∂Ω is of C 2 and H(x) at x ∈ ∂Ω nonpositive with respect to the ouward normal.We have from (