Optimal results for the fractional heat equation involving the Hardy potential

TL;DR

This paper investigates the fractional heat equation with Hardy potential, establishing optimal existence and blow-up results, and identifying a critical exponent that determines solution behavior.

Contribution

It provides the first optimal existence and blow-up criteria for the fractional heat equation with Hardy potential, including a critical exponent for semilinear cases.

Findings

Threshold for existence based on Hardy inequality constant

Complete blow-up occurs for certain parameters

Existence of a critical power p_+(s,λ) for solutions

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}