Fujita exponent for non-local parabolic equation involving the 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

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

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.