On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity

In this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation ut − mu +a(x)|u|αu = f0(u) Ω K(y)|u(y, t)|β dy +g(x) in Ω ×(0,∞) with the initial data u(x, 0) = u0(x) ∈ Lq , q 1, and zero boundary condition in ∂Ω. Two cases for a(x) a0 > 0 and a(x) 0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p max{1, q}. The attractor A is in fact a bounded set in W 1,m 0 ∩ L∞ if a(x) a0 > 0 in Ω, and A is bounded in W 1,m 0 ∩ Lp if a(x) 0 in Ω. © 2007 Elsevier Inc. All rights reserved