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 $${\partial }_{t}u=\Delta u^m+|x{|}^{-2}{u}^p,(x,t)\in {\Re }^N\times (0,\infty ),$$ in the range of exponents $1\le p<m$ and dimension $N\ge 3$. The self-similar solution is unbounded at $x=0$ and has a logarithmic vertical asymptote, but it remains bounded at any $x\ne 0$ and $t\in (0,\infty )$ and it is a weak solution in $L^1$ sense, which moreover satisfies $u(t)\in L^p({\Re }^N)$ for any $t>0$ and $p\in [1,\infty )$. 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 $u_0$, contrasting with previous results in literature for the critical limit $p=m$.