Asymptotic behavior of a degenerate nonlocal parabolic equation

In this paper, we consider the asymptotic behavior for the degenerate nonlocal parabolic equation u t = ∇ ⋅ ( u 3 ∇ u ) + λ f ( u ) ( ∫ Ω f ( u ) d x ) 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 0 < p ⩽ 1 , u ( x , t ) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0 ; (b) for 1 < p < 2 , u ( x , t ) is globally bounded for any λ > 0 , 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 u 0 ( 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 .