Global attractors for p-Laplacian equation

Abstract The existence of a ( L 2 ( Ω ) , W 0 1 , p ( Ω ) ∩ L q ( Ω ) ) -global attractor is proved for the p-Laplacian equation u t − div ( | ∇ u | p − 2 ∇ u ) + f ( u ) = g on a bounded domain Ω ⊂ R n ( n ⩾ 3 ) with Dirichlet boundary condition, where p ⩾ 2 . The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c 1 | u | q − k ⩽ f ( u ) u ⩽ c 2 | 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.