Global attractors for p-Laplacian equation

Abstract

The existence of a (L2 (Ω), W01, p (Ω) ∩ Lq (Ω))-global attractor is proved for the p-Laplacian equation ut - div (| ∇ u |p - 2 ∇ u) + f (u) = g on a bounded domain Ω ⊂ Rn(n ≥ 3) with Dirichlet boundary condition, where p ≥ 2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c1 | u |q - k ≤ f (u) u ≤ c2 | u |q + k and f′ (u) ≥ - l, where q ≥ 2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate. © 2006 Elsevier Inc. All rights reserved.