Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

We study the property of finite time vanishing of solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations u t − ∑ i = 1 n D i ( a i ( x , t , u ) | D i u | p i ( x , t ) − 2 D i u ) + c ( x , t ) | u | σ ( x , t ) − 2 u = f ( x , t ) with variable exponents of nonlinearity p i ( x , t ) , σ ( x , t ) ∈ ( 1 , ∞ ) . We show that the solutions of this problem may vanish in a finite time even if the equation combines the directions of slow and fast diffusion and estimate the extinction moment in terms of the data. If the solution does not identically vanish in a finite time, we estimate the rate of vanishing of the solution as t → ∞ . We establish conditions on the nonlinearity exponents which guarantee vanishing of the solution at a finite instant even if the equation eventually transforms into the linear one.