This paper investigates the (fractional) heat equation with a nonlocal nonlinearity involving a Riesz potential: \(\begin{aligned} u_{t}+(-\Delta )^{\frac{\beta }{2}} u= I_\alpha (|u|^{p}),\qquad x\in \mathbb {R}^n,\,\,\,t>0, \end{aligned}\) where \(\alpha \in (0,n)\) , \(\beta \in (0,2]\) , \(n\ge 1\) , \(p>1.\) We introduce the Fujita-type critical exponent \(p_{\textrm{Fuj}}(n,\beta ,\alpha )=1+(\beta +\alpha )/(n-\alpha )\) , which characterizes the global behavior of solutions: global existence for small initial data when \(p>p_{\textrm{Fuj}}(n,\beta ,\alpha ),\) and finite-time blow-up when \(p\le p_{\textrm{Fuj}}(n,\beta ,\alpha )\) . It is remarkable that the critical Fujita exponent is not determined by the usual scaling argument that yields \(p_{sc}=1+(\beta +\alpha )/n\) , but instead arises in an unconventional manner, similar to the results of Cazenave et al. [Nonlinear Analysis, 68 (2008), 862-874] for the heat equation with a nonlocal nonlinearity of the form \(\int _0^t(t-s)^{-\gamma }|u(s)|^{p-1}u(s)ds,\,0\le \gamma <1.\) The result on global existence for \(p>p_{\textrm{Fuj}}(n,2,\alpha ),\) provides a positive answer to the hypothesis proposed by Mitidieri and Pohozaev in [Proc. Steklov Inst. Math., 248 (2005) 164???185]. We further establish global nonexistence results for the above heat equation, where the Riesz potential term \(I_\alpha (|u|^{p})\) is replaced by a more general convolution operator \((\mathcal {K}*|u|^p),\,\mathcal {K}\in L^1_{loc}\) , thereby extending the Mitidieri???Pohozaev’s results established in the aforementioned work. Proofs of the blow-up results are obtained using a nonlinear capacity method specifically adapted to the structure of the problem, while global existence is established via a fixed-point argument combined with the Hardy???Littlewood???Sobolev inequality.
