A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions

We study the localization properties of weak solutions to the Dirichlet problem for the degenerate parabolic equationut-div(|u|γ(x,t)∇u)=f,with variable exponent of nonlinearity γ. We prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent γ(x,t) sufficient for the existence of such properties as finite speed of propagation of disturbances, the waiting time effect, finite time vanishing of the solution. It is shown that the solution may instinct in a finite time even if γ≡γ(x)⩽0 in the problem domain but maxγ=0.