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}+(-\Delta)^{\frac{\beta}{2}} u= I_\alpha(|u|^{p}),\qquad x\in \mathbb{R}^n,\,\,\,t>0, \end{equation*} where $\alpha\in(0,n)$, $\beta\in(0,2]$, $n\geq1$, $p>1.$ We introduce the Fujita-type critical exponent $p_{\mathrm{Fuj}}(n,\beta,\alpha)=1+(\beta+\alpha)/(n-\alpha)$, which characterizes the global behavior of solutions: global existence for small initial data when $p>p_{\mathrm{Fuj}}(n,\beta,\alpha),$ and finite-time blow-up when $p\leq p_{\mathrm{Fuj}}(n,\beta,\alpha)$. 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 $\int_0^t(t-s)^{-\gamma}|u(s)|^{p-1}u(s)ds,\,0\leq \gamma<1.$ The result on global existence for $p>p_{\mathrm{Fuj}}(n,2,\alpha),$ 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 $(\mathcal{K}\ast |u|^p),\,\mathcal{K}\in L^1_{loc}$, 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.