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

TL;DR

This paper constructs a unique self-similar solution for a reaction-diffusion equation with Hardy potential, demonstrating existence of global solutions for a range of exponents and initial conditions, contrasting prior results.

Contribution

It establishes the existence and uniqueness of a specific self-similar solution with Hardy potential and proves global solutions for bounded, compactly supported initial data.

Findings

Existence of a self-similar solution with logarithmic asymptote at the origin.

Global existence of solutions for initial data with bounded support.

Contrasts with previous results in the critical case p=m.

Abstract

Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=\Delta u^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in the range of exponents $1\leq p<m$ and dimension $N\geq3$. The self-similar solution is unbounded at $x=0$ and has a logarithmic vertical asymptote, but it remains bounded at any $x\neq0$ and $t\in(0,\infty)$ and it is a weak solution in $L^1$ sense, which moreover satisfies $u(t)\in L^p(\real^N)$ 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 $p=m$.