Asymptotic behavior of a degenerate nonlocal parabolic equation

In this paper, we consider the asymptotic behavior for the degenerate nonlocal parabolic equationut=∇⋅(u3∇u)+λf(u)(∫Ωf(u)dx)p,x∈Ω,t>0, with a homogeneous Dirichlet boundary condition, where λ>0, p>0 and f is decreasing. It is found that (a) for 00; (b) for 10, moreover, if Ω is a ball, the stationary solution is unique and globally asymptotically stable; (c) for p=2, if 0<λ<2|∂Ω|2, then u(x,t) is globally bounded, moreover, if Ω is a ball, the stationary solution is unique and globally asymptotically stable; if λ>2|∂Ω|2, there is no stationary solution and u(x,t) blows up in finite time for all x∈Ω; (d) for p>2, there exists a λ∗>0 such that for λ>λ∗, or for 0<λ⩽λ∗ and u0(x) sufficiently large, u(x,t) blows up in finite time for all x∈Ω. Moreover, some formal asymptotic estimates for the behavior of u(x,t) as it blows up are obtained for p⩾2.