Fractional heat equation involving Hardy-Leray Potential

Abstract

In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem $$ \begin{cases} (w_t-\Delta w)^s=\frac{\lambda}{|x|^{2s}} w+w^p +f, &\text{ in }\mathbb{R}^N\times (0,+\infty),\\ w(x,t)=0, &\text{ in }\mathbb{R}^N\times (-\infty,0], \end{cases} $$ where $N> 2s$, $0<s<1$ and $0<\lambda<\Lambda_{N,s}$, the optimal constant in the fractional Hardy-Leray inequality. In particular we show the existence of a critical existence exponent $p_{+}(\lambda, s)$ and of a Fujita-type exponent $F(\lambda,s)$ such that the following holds: - Let $p>p_+(\lambda,s)$. Then there are not any non-negative supersolutions. - Let $p<p_+(\lambda,s)$. Then there exist local solutions while concerning global solutions we need to distinguish two cases: - Let $ 1< p\le F(\lambda,s)$. Here we show that a weighted norm of any positive solution blows up in finite time. - Let $F(\lambda,s)<p<p_+(\lambda,s)$. Here we prove the existence of global solutions under suitable hypotheses.