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

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

2010-09-23

arxiv.org/abs/1009.4591v1

Mathematics

Analysis of PDEs

Scientific paper

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 $p<1+\frac{2(2+\alpha)}{N+2+\sqrt{(N-2)^2-4\kappa}}$.

Affiliated with

Also associated with

No associations

Rating

Singular solutions to the heat equations with nonlinear absorption and Hardy potentials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Singular solutions to the heat equations with nonlinear absorption and Hardy potentials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Singular solutions to the heat equations with nonlinear absorption and Hardy potentials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-338705