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

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 ${\mathbb{R}}^N\times [0,\infty )$, $N\ge 3$, with a singularity at the point $(0,0)$, that is, nonnegative solutions satisfying $u(x,0)=0$ for $x\ne 0$, 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 $p<1+\frac{2(2+\alpha )}{N+2+\sqrt{(N-2{)}^2-4\kappa }}$.