Singular solutions to the heat equations with nonlinear absorption and Hardy potentials

TL;DR

This paper classifies singular solutions to a heat equation with nonlinear absorption and Hardy potential, identifying conditions for existence and uniqueness based on parameters, by transforming the problem into a weighted Laplace-Beltrami operator framework.

Contribution

It provides a complete classification of singular solutions for the heat equation with Hardy potential and nonlinear absorption, including existence, nonexistence, and uniqueness results.

Findings

Singular solutions exist if and only if p<1+2(2+α)/(N+2+√((N-2)^2-4κ))

The problem is transformed into a weighted Laplace-Beltrami operator setting.

Classification includes source solutions and very singular solutions.

Abstract

We study the existence and nonexistence of singular solutions to the equation $u_t-\Delta u - \frac{\kappa}{|x|^2}u+|x|^\alpha u|u|^{p-1}=0$, $p>1$, in $\R^N\times[0,\infty)$, $N\ge 3$, with a singularity at the point $(0,0)$, that is, nonnegative solutions satisfying $u(x,0)=0$ for $x\ne0$, assuming that $\a>-2$ and $\kappa<\left(\frac{N-2}2\right)^2$. The problem is transferred to the one for a weighted Laplace-Beltrami operator with a non-linear absorbtion, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a {\it source solution} with a multiple of the Dirac mass as initial datum, or as a unique {\it very singular solution}, leads to a complete classification of singular solutions to the original problem, which exist if and only if $p<1+\frac{2(2+\alpha)}{N+2+\sqrt{(N-2)^2-4\kappa}}$.