Localization for a Porous Medium Type Equation in High Dimensions

We prove the strict localization for a porous medium type equation with a source term, ut = ∇(uσ∇u)+uβ , x ∈ RN , N > 1, β > σ+ 1, σ > 0, in the case of arbitrary compactly supported initial functions u0. We also otain an estimate of the size of the localization in terms of the support of the initial data supp u0 and the blow-up time T . Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.