Existence and Non-existence of Global Solutions for a Nonlocal Choquard–Kirchhoff Diffusion Equations in $\mathbb {R}^{N}$

Abstract

In this paper, we investigate the local existence, global existence, and blow-up of solutions to the Cauchy problem for Choquard–Kirchhoff-type equations involving the fractional p-Laplacian. As a particular case, we study the following initial value problem

$$\begin{aligned}\left\{ \begin{array}{llc} u_{t}+M\left( \Vert u\Vert ^{p}\right) [(-\Delta )^{s}_{p}u+V(x)|u|^{p-2}u]=\left( \int _{\mathbb {R}^{N}}\frac{|u|^{q}}{|x-y|^{\mu }}\,dy\right) |u|^{q-2} u&{} \text {in}\ &{}\mathbb {R}^{N}\times (0, +\infty ), \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\mathbb {R}^{N} , \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} \Vert u\Vert =\left( [u]^{p}_{s,p}+\int _{\mathbb {R}^{N}}V(x)|u|^{p}\,dx\right) ^{1/p}, \end{aligned}$$

$s\in (0,1)$, $N>ps$, $p,q> 2$, $(-\Delta )^{s}_{p}$ is the fractional p-Laplacian, $u_{0} :\mathbb {R}^{N}\rightarrow [0, +\infty )$ is the initial function, $M :\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ is a continuous function given by $M(\sigma )=\sigma ^{\theta -1}$, $\theta \in [1, N/(N-sp))$ and $V :\mathbb {R}^{N}\rightarrow \mathbb {R}^{+}$ is the potential function. Under some appropriate conditions, the well-posedness of nonnegative solutions for the above Cauchy problem is established by employing the Galerkin method. Moreover, the asymptotic behavior of global solutions is investigated under some assumptions on the initial data. We also establish upper and lower bounds for the blow-up time.