Focusing solutions of porous medium equations with reaction

In this paper we study the existence of focusing solution to a class of porous medium equations taking the form ut=Δum+F(x,u,∇u), where m>1. Focusing solution has the property that its initial distribution is in the exterior of a finite domain. That is, there is a hole in the support of initial value, and in finite time T the hole disappears. We show there exists a focusing solution for a number of important models in physics and biology. Such solution is an example of a self-similar solution of the second kind. That is, the similarity variables cannot be determined a priori from dimensional consideration. Furthermore, it serves the purpose of supplying concrete bounds for the optimal regularity of general solutions of the equation. The P-Laplacian counterpart of this equation is also studied.