Optimal results for the fractional heat equation involving the Hardy potential

Abstract

In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem $$(P_\theta)\quad \left\{ \begin{array}{rcl} u_t+(-\Delta)^{s} u&=&\l\dfrac{\,u}{|x|^{2s}}+\theta u^p+ c f\mbox{ in } \Omega\times (0,T),\\ u(x,t)&>&0\inn \Omega\times (0,T),\\ u(x,t)&=&0\inn (\ren\setminus\Omega)\times[ 0,T),\\ u(x,0)&=&u_0(x) \mbox{ if }x\in\O, \end{array} \right. $$ where $N> 2s$, $0<s<1$, $(-\Delta)^s$ is the fractional Laplacian of order $2s$, $p>1$, $c,\l>0$, $u_0\ge 0$, $f\ge 0$ are in a suitable class of functions and $\theta=\{0,1\}$. Notice that $(P_0)$ is a linear problem, while $(P_1)$ is a semilinear problem. The main features in the article are: \begin{enumerate} \item Optimal results about \emph{existence} and \emph{instantaneous and complete blow up} in the linear problem $(P_0)$, where the best constant $\Lambda_{N,s}$ in the fractional Hardy inequality provides the threshold between existence and nonexistence. Similar results in the local heat equation were obtained by Baras and Goldstein in \cite{BaGo}. However, in the fractional setting the arguments are much more involved and they require the proof of a weak Harnack inequality for a weighted operator that appear in a natural way. Once this Harnack inequality is obtained, the optimal results follow as a simpler consequence than in the classical case. \item The existence of a critical power $p_+(s,\lambda)$ in the semilinear problem $(P_1)$ such that: \begin{enumerate} \item If $p> p_+(s,\lambda)$, the problem has no weak positive supersolutions and a phenomenon of \emph{complete and instantaneous blow up} happens. \item If $p< p_+(s,\lambda)$, there exists a positive solution for a suitable class of nonnegative data. \end{enumerate} \end{enumerate}