Existence of Global Attractors in for -Laplacian Parabolic Equation in

Caisheng Chen,Lanfang Shi,Hui Wang

doi:10.1155/2009/563767

Caisheng Chen, Lanfang Shi + Show 1 more

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https://doi.org/10.1155/2009/563767

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Abstract

We study the long-time behavior of solution for the -Laplacian equation in , in which the nonlinear term is a function like with , , or with and . We prove the existence of a global -attractor for any .

Highlights

In this paper we are interested in the existence of a global L2 RN, Lp RN -attractor for the m-Laplacian equation ut − Δmu λ|u|m−2u f x, u g x, x ∈ RN, t ∈ R, 1.1 with initial data condition u x, 0 u0 x, x ∈ RN, 1.2 where the m-Laplacian operator Δmu div |∇u|m−2∇u, 2 ≤ m < N, λ > 0
For the case m 2, the existence of global L2 RN, L2 RN -attractor for 1.1 - 1.2 is proved by Wang in 1 under appropriate assumptions on f and g
Khanmamedov 2 studied the existence of global L2 RN, Lm∗ RN -attractor for 1.1 - 1.2 with m∗ mN/ N −m

Summary

Introduction

In this paper we are interested in the existence of a global L2 RN , Lp RN -attractor for the m-Laplacian equation ut − Δmu λ|u|m−2u f x, u g x , x ∈ RN, t ∈ R , 1.1 with initial data condition u x, 0 u0 x , x ∈ RN, 1.2 where the m-Laplacian operator Δmu div |∇u|m−2∇u , 2 ≤ m < N, λ > 0. We derive L∞ estimate of solutions by Moser’s technique as in 5–7 , and due to this, we need not to make the assumption like fu x, u ≥ a4 x to show the uniqueness. The nonlinear function f x, u −h x |u|q−2u with h x ≥ 0, q ≥ 1 does not satisfy the assumption 1.3. In this paper, motivated by 2–4 , we are interested in the global L2 RN , Lp RN - attractor Ap for the problem 1.1 - 1.2 with any p > m, in which p is independent of the order of polynomial for u on f x, u.

Preliminaries

Global Attractor in RN