A special self-similar solution and existence of global solutions for a reaction-diffusion equation with Hardy potential

Abstract

Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential∂tu=Δum+|x|−2up,(x,t)∈RN×(0,∞), in the range of exponents 1≤p<m and dimension N≥3. The self-similar solution is unbounded at x=0 and has a logarithmic vertical asymptote, but it remains bounded at any x≠0 and t∈(0,∞) and it is a weak solution in L1 sense, which moreover satisfies u(t)∈Lp(RN) for any t>0 and p∈[1,∞). As an application of this self-similar solution, it is shown that there exists at least a weak solution to the Cauchy problem associated to the previous equation for any bounded, nonnegative and compactly supported initial condition u0, contrasting with previous results in literature for the critical limit p=m.