Fujita exponent for non-local parabolic equation involving the Hardy–Leray potential

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 \(\begin{aligned} {\left\{ \begin{array}{ll} (w_t-\Delta w)^s=\frac{\lambda }{|x|^{2s}} w+w^p +f, &{}\quad \text {in }\mathbb {R}^N\times (0,+\infty ),\\ w(x,t)=0, &{}\quad \text {in }\mathbb {R}^N\times (-\infty ,0], \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(w_t-\mathrm{\Delta }w)}^s=\frac{\lambda }{{|x|}^{2s}}w+w^p+f,\hfill & \text{in}{\mathbb{R}}^N\times (0,+\infty ),\hfill \\ w(x,t)=0,\hfill & \text{in}{\mathbb{R}}^N\times (-\infty ,0],\hfill \end{array}\right)\end{array}$$ where \(N> 2s\) $N>2s$ , \(0<s<1\) $0<s<1$ and \(0<\lambda <\Lambda _{N,s}\) $0<\lambda <{\mathrm{\Lambda }}_{N,s}$ , the optimal constant in the fractional Hardy–Leray inequality. In particular, we show the existence of a critical existence exponent \(p_{+}(\lambda , s)\) $p_{+}(\lambda ,s)$ and of a Fujita-type exponent \(F(\lambda ,s)\) $F(\lambda ,s)$ such that the following holds:

Let \(p>p_+(\lambda ,s)\) $p>p_{+}(\lambda ,s)$ . Then there are not any non-negative supersolutions.