Parabolic problems whose Fujita critical exponent is not given by scaling

Abstract

This paper investigates the (fractional) heat equation with a nonlocal nonlinearity involving a Riesz potential: \begin{equation*} u_{t}+(-Δ)^{\fracβ{2}} u= I_α(|u|^{p}),\qquad x\in \mathbb{R}^n,\,\,\,t>0, \end{equation*} where $\hat{I}\pm \in (0,n)$, ${\hat{I}}^2\in (0,2]$, $n\ge 1$, $p>1.$ We introduce the Fujita-type critical exponent $p_{\mathrm{Fuj}}(n,{\hat{I}}^2,\hat{I}\pm )=1+({\hat{I}}^2+\hat{I}\pm )/(n-\hat{I}\pm )$, which characterizes the global behavior of solutions: global existence for small initial data when $p>p_{\mathrm{Fuj}}(n,{\hat{I}}^2,\hat{I}\pm ),$ and finite-time blow-up when $p\le p_{\mathrm{Fuj}}(n,{\hat{I}}^2,\hat{I}\pm )$. It is remarkable that the critical Fujita exponent is not determined by the usual scaling argument that yields $p_{sc}=1+({\hat{I}}^2+\hat{I}\pm )/n$, 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 ${\int }_0^t(t-s{)}^{-{\hat{I}}^3}|u(s){|}^{p-1}u(s)ds,0\le {\hat{I}}^3<1.$ The result on global existence for $p>p_{\mathrm{Fuj}}(n,2,\hat{I}\pm ),$ 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 $I_{\hat{I}}\pm (|u{|}^p)$ is replaced by a more general convolution operator $(\mathcal{K}\ast |u{|}^p),\mathcal{K}\in L_{loc}^1$, 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.

22 pages, updated version. arXiv admin note: substantial text overlap with arXiv:2510.11648