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

We study the behavior of energy solutions of the homogeneous Dirichlet problem for the anisotropic doubly degenerate parabolic equation ddt(|v|m(x,t)signv)=∑i=1nDi(ai(x,t)|Div|pi(x,t)−2Div)+b(x,t)|v|σ(x,t)−2v+g(x,t). The exponents of nonlinearity m(x,t)>0m(x,t)>0, pi(x,t)>1pi(x,t)>1 and σ(x,t)>1σ(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→∞t→∞. It is shown that the possibility of the finite time blow-up or extinction depends on the properties of mtmt 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=0b=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)−1v+b(x,t)|v|σ(x,t)−2vc(x,t)|v|m(x,t)−1v+b(x,t)|v|σ(x,t)−2v, and to the equations, which transform into linear equations as t→∞t→∞.