Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential

Abstract

We prove existence and uniqueness of self-similar solutions with exponential form u(x,t)=e^{alpha t}f(|x|e^{-beta t}), alpha, beta>0, to the quasilinear reaction-diffusion equation \partial_t u=Delta u^m+|x|^{sigma}u^p, with m>1, 1<p<m and sigma=-2(p-1)/(m-1). Such self-similar solutions are usually known in the literature as eternal solutions since they exist for any t\in(-\infty,\infty). As an application of the existence of these eternal solutions, we show existence of global in time weak solutions with any initial condition u_0 in L^{\infty}(R^N) and, in particular, that these weak solutions remain compactly supported at any time t>0 if u_0 is compactly supported.