Doubly degenerate parabolic equations with variable nonlinearity II: Blow-up and extinction in a finite time

摘要

We study the behavior of energy solutions of the homogeneous Dirichlet problem for the anisotropic doubly degenerate parabolic equation d/(dt)(|v|~(m(x,t)) sign v) = ~n∑_(i=1)D_i(a_i(x, t)|D_iv|~(p_i(x,t)?2)D_iv)+b(x, t)|v|~(σ(x,t)?2)v + g(x, t). The exponents of nonlinearitym(x, t) > 0, p_i(x, t) > 1 and σ(x, t) > 1 are given functions. We derive sufficient conditions of the finite time blow-up or vanishing and establish the decay rates as t → ∞. It is shown that the possibility of the finite time blow-up or extinction depends on the properties of mt and that the anisotropy of the diffusion part of the equation may cause extinction in a finite time even in the absence of the absorption term (b = 0). The results concerning the finite-time extinction are extended to the equations with the low-order terms of critical growth, c(x, t)|v|~(m(x,t)?1)v + b(x, t)