Fujita-type results for parabolic equations with Hartree-type nonlinearities

TL;DR

This work analyzes global-in-time behavior for a parabolic equation with a Hartree-type nonlinearity $u_t + (-\Delta)^{\frac{\beta}{2}}u = (\mathcal{K} * |u|^p)|u|^q$ on $\mathbb{R}^n$, spanning local existence, global existence, and Fujita-type nonexistence/blow-up phenomena. The authors combine nonlinear capacity methods, fixed-point arguments, and Hardy–Littlewood–Sobolev inequalities to treat general kernels $\mathcal{K}$ and the Riesz kernel, deriving sharp thresholds that separate global existence from finite-time blow-up. For kernels with suitable growth, global nonexistence results are proved when $p+q>2$ (or under a decay condition on initial data), while global existence is obtained for Riesz-type kernels under small initial data in the scaling-critical space with exponent $q_{sc}=\dfrac{n(p+q-1)}{\beta+\alpha}$. The results extend and refine prior work by Filippucci and Ghergu, clarifying the roles of nonlocal Hartree nonlinearities and fractional diffusion in Fujita-type phenomena and offering a precise set of critical exponents and function-space conditions governing global behavior.

Abstract

This paper investigates the critical behavior of global solutions to a parabolic equation with a Hartree-type nonlinearity of the form $$\left\{\begin{array}{ll} u_{t}+(-Δ)^{\fracβ{2}} u= (\mathcal{K}\ast |u|^{p})|u|^{q},&\qquad x\in \mathbb{R}^n,\,\,\,t>0, u(x,0)=u_{0}(x),& \qquad x\in \mathbb{R}^n,\end{array} \right.$$ where $β\in(0,2]$, $n\geq1$, $p>1$, $q\geq 1$, $(-Δ)^{\fracβ{2}},\,β\in(0,2)$ denotes the fractional Laplacian, the symbol $\ast$ denotes the convolution operation in $\mathbb{R}^n$, and $\mathcal{K}:(0,\infty)\rightarrow(0,\infty)$ is a continuous function such that $\mathcal{K}(|\cdotp|)\in L^1_{loc}(\mathbb{R}^n)$ and is monotonically decreasing in a neighborhood of infinity. We establish conditions for the global nonexistence of solutions to the problem under consideration, thereby partially improving some results of Filippucci and Ghergu in [Discrete Contin. Dyn. Syst. A, 42 (2022) 1817-1833] and [Nonlinear Anal., 221 (2022) 112881]. In addition, we establish local and global existence results in the case where the convolution term corresponds to the Riesz potential. Our methodology relies on the nonlinear capacity method and the fixed-point principle, combined with the Hardy-Littlewood-Sobolev inequality.

Theorems & Definitions (17)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Definition 1.6
• Theorem 1.7: Local existence
• Theorem 1.8: Global existence
• Theorem 1.9: Blow-up
• Remark 1.10