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 u(t) - Delta(m)u + a(x)vertical bar u vertical bar(alpha)u = f(0)(u) integral(Omega) K(y)vertical bar u(y, t)vertical bar(beta) dy + g(x) in Omega x (0, infinity) with the initial data u(x, 0) = u(0)(x) is an element of L-q, q >= 1, and zero boundary condition in partial derivative Omega. Two cases for a (x) >= a(0) > 0 and (x) >= 0 are considered. We obtain the existence and L-P estimate of global attractor A in L-p, for any p >= max {1, q}. The attractor A is in fact a bounded set in W-0(1,m) boolean AND L-infinity if a(x) >= a(0) > 0 in Omega, and A is bounder in W-0(1,m) boolean AND L-p if a(x) >= 0 in Omega. (C) 2007 Elsevier Inc. All rights reserved.