Localization for a porous medium type equation in high dimensions

Abstract

We prove the strict localization for a porous medium type equation with a source term, $u_{t}= \nabla (u^ {\sigma } \nabla u)+u^ \beta$, $x \in \mathbf {R}^ N$, $N>1$, $\beta >\sigma +1$, $\sigma >0,$ in the case of arbitrary compactly supported initial functions $u_0$. We also otain an estimate of the size of the localization in terms of the support of the initial data $\operatorname {supp}u_0$ 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.