Asymptotic behavior of a degenerate nonlocal parabolic equation

In this paper, we consider the asymptotic behavior for the degenerate nonlocal parabolic equation u(t) = del.(u(3)del u) + lambda f(u)/(integral(Omega) f(u)dx)(p), x is an element of Omega, t > 0, with a homogeneous Dirichlet boundary condition, where lambda > 0, p > 0 and f is decreasing. It is found that (a) for 0 < p <= 1, u(x, t) is globally bounded and the unique stationary solution is globally asymptotically stable for any lambda > 0; (b) for 1 < p < 2, u(x, t) is globally bounded for any lambda > 0, moreover, if Omega is a ball, the stationary solution is unique and globally asymptotically stable: (c) for p = 2, if 0 < lambda < 2 vertical bar partial derivative Omega vertical bar(2), then it (x, t) is globally bounded, moreover, if Omega is a ball. the stationary solution is unique and globally asymptotically stable; if lambda > 2 vertical bar partial derivative Omega vertical bar(2), there is no stationary solution and u(x, t) blows up in finite time for all x epsilon Omega; (d) for p > 2, there exists a lambda* > 0 such that for lambda > lambda*, or for 0 < lambda <= lambda* and u(0)(x) sufficiently large, u(x, t) blows up in finite time for all x is an element of Omega. Moreover, some formal asymptotic estimates for the behavior of u(x, t) as it blows up are obtained for p >= 2. (C) 2009 Elsevier Inc. All rights reserved.