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 $$\{\begin{array}{ll}(w_t-\Delta w{)}^s=\frac{\lambda }{|x{|}^{2s}}w+w^p+f,& \text{ in }{\mathbb{R}}^N\times (0,+\infty ),\\ w(x,t)=0,& \text{ in }{\mathbb{R}}^N\times (-\infty ,0],\end{array}$$ 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.