A note on the Fujita exponent in Fractional heat equation involving the Hardy potential

TL;DR

This paper investigates the Fujita exponent and blow-up phenomena for a fractional heat equation with Hardy potential, extending understanding of critical exponents in fractional PDEs with singular potentials.

Contribution

It characterizes the Fujita exponent and blow-up behavior for the fractional heat equation involving Hardy potential, clarifying the critical exponent range.

Findings

Determined the Fujita exponent for the fractional heat equation with Hardy potential.

Established conditions for blow-up versus global existence.

Extended previous results to the fractional setting with singular potential.

Abstract

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the Fractional Cauchy problem with the Hardy potential, namely, \begin{equation*} u_t+(-\Delta)^s u=\lambda\dfrac{u}{|x|^{2s}}+u^{p}\inn\ren,\\ u(x,0)=u_{0}(x)\inn\ren, \end{equation*} where $N> 2s$, $0<s<1$, $(-\Delta)^s$ is the fractional laplacian of order $2s$, $\l>0$, $u_0\ge 0$, and $1<p<p_{+}(s,\lambda)$, where $p_{+}(\lambda, s)$ is the critical existence power found in \cite{BMP} and \cite{AMPP}.