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

Abstract

AbstractIn 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 $$\begin{aligned} {\left\{ \begin{array}{ll} (w_t-\Delta w)^s=\frac{\lambda }{|x|^{2s}} w+w^p +f, &{}\quad \text {in }\mathbb {R}^N\times (0,+\infty ),\\ w(x,t)=0, &{}\quad \text {in }\mathbb {R}^N\times (-\infty ,0], \end{array}\right. } \end{aligned}$$ ( w t - Δ w ) s = λ | x | 2 s w + w p + f , in R N × ( 0 , + ∞ ) , w ( x , t ) = 0 , in R N × ( - ∞ , 0 ] , where $$N> 2s$$ N > 2 s , $$0<s<1$$ 0 < s < 1 and $$0<\lambda <\Lambda _{N,s}$$ 0 < λ < Λ 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 + ( λ , s ) and of a Fujita-type exponent $$F(\lambda ,s)$$ F ( λ , s ) such that the following holds: Let $$p>p_+(\lambda ,s)$$ p > p + ( λ , s ) . Then there are not any non-negative supersolutions. Let $$p<p_+(\lambda ,s)$$ p < p + ( λ , s ) . Then there exist local solutions, while concerning global solutions we need to distinguish two cases: Let $$ 1< p\le F(\lambda ,s)$$ 1 < p ≤ F ( λ , s ) . Here we show that a weighted norm of any positive solution blows up in finite time. Let $$F(\lambda ,s)<p<p_+(\lambda ,s)$$ F ( λ , s ) < p < p + ( λ , s ) . Here we prove the existence of global solutions under suitable hypotheses.