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

We study the existence and nonexistence of singular solutions to the equation $u_t-Δu - \fracκ{|x|^2}u+|x|^αu|u|^{p-1}=0$, $p>1$, 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 $κ<\left(\frac{N-2}2\right)^2$. 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 $p<1+\frac{2(2+α)}{N+2+\sqrt{(N-2)^2-4κ}}$.

No reviews yet.

No comments yet.