Parabolic problems whose Fujita critical exponent is not given by scaling
Ahmad Z. Fino, Berikbol T. Torebek
TL;DR
The article identifies a Fujita-type critical exponent $p_{\mathrm{Fuj}}(n,\beta,\alpha)=1+\frac{\beta+\alpha}{n-\alpha}$ for the parabolic equation with a fractional Laplacian and nonlocal nonlinearity $I_\alpha(|u|^p)$, showing global existence for small data when $p>p_{\mathrm{Fuj}}$ and finite-time blow-up for $p\le p_{\mathrm{Fuj}}$, with this threshold not governed by scaling. It connects to nonlocal nonlinearities and extends classical results by demonstrating global existence for $p>p_{\mathrm{Fuj}}(n,2,\alpha)$ and to general convolution kernels, using nonlinear capacity methods for blow-up and fixed-point techniques with Hardy–Littlewood–Sobolev inequalities for global existence. The work also confirms Mitidieri–Pohozaev's conjecture in the special case $\beta=2$ and broadens nonexistence results to kernels beyond the Riesz potential, highlighting a novel critical exponent arising from nonlocality rather than scaling. Overall, the paper advances understanding of nonlocal parabolic problems and provides rigorous criteria for global behavior versus blow-up across fractional and convolutive nonlinearities.
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 $α\in(0,n)$, $β\in(0,2]$, $n\geq1$, $p>1.$ We introduce the Fujita-type critical exponent $p_{\mathrm{Fuj}}(n,β,α)=1+(β+α)/(n-α)$, which characterizes the global behavior of solutions: global existence for small initial data when $p>p_{\mathrm{Fuj}}(n,β,α),$ and finite-time blow-up when $p\leq p_{\mathrm{Fuj}}(n,β,α)$. It is remarkable that the critical Fujita exponent is not determined by the usual scaling argument that yields $p_{sc}=1+(β+α)/n$, 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 $\int_0^t(t-s)^{-γ}|u(s)|^{p-1}u(s)ds,\,0\leq γ<1.$ The result on global existence for $p>p_{\mathrm{Fuj}}(n,2,α),$ 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 $I_α(|u|^{p})$ 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.