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

Abstract

This paper investigates the critical behavior of global solutions to a parabolic equation with a Hartree-type nonlinearity of the form $$\left\{\begin{array}{ll} u_{t}+(-\Delta)^{\frac{\beta}{2}} u= (\mathcal{K}\ast |u|^{p})|u|^{q},&\qquad x\in \mathbb{R}^n,\,\,\,t>0, u(x,0)=u_{0}(x),& \qquad x\in \mathbb{R}^n,\end{array} \right.$$ where $\beta\in(0,2]$, $n\geq1$, $p>1$, $q\geq 1$, $(-\Delta)^{\frac{\beta}{2}},\,\beta\in(0,2)$ 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.