Global behavior of nonlocal in time reaction-diffusion equations

TL;DR

This paper analyzes a nonlocal-in-time reaction-diffusion model $\partial_t (k\ast(u-u_0))+\mathcal{L}_x[u]=f(u)$ on $\Omega$ with Sonine kernel assumptions, establishing local and global strong solutions, $L^2$-decay rates, and finite-time blow-up criteria. The approach blends energy estimates, comparison principles, Jensen's inequality for convex nonlinearities, and Volterra-type integral inequalities to derive sharp bounds such as $\|u(t,\cdot)\|_{L^2}\le \|u_0\|_{L^2}/(1+C_{\mathcal{L}}(l\ast 1)(t))$, and to identify blow-up thresholds tied to the initial mass $\int_\Omega u_0\phi_1$. The work unifies a broad class of time-memory kernels (fractional, distributed-order, tempered, etc.) and connects to open questions of Gal–Warma and Luchko–Yamamoto, while also outlining extensions to quasilinear diffusion operators and highlighting remaining open questions on decay and blow-up in that setting. These results advance the mathematical understanding of anomalous and ultraslow diffusion phenomena with memory in nonlinear media.

Abstract

The present paper considers the Cauchy-Dirichlet problem for the time-nonlocal reaction-diffusion equation $$\partial_t (k\ast(u-u_0))+\mathcal{L}_x [u]=f(u),\,\,\,\, x\inΩ\subset\mathbb{R}^n, t>0,$$ where $k\in L^1_{loc}(\mathbb{R}_+),$ $f$ is a locally Lipschitz function, $\mathcal{L}_x$ is a linear operator. This model arises when studying the processes of anomalous and ultraslow diffusions. Results regarding the local and global existence, decay estimates, and blow-up of solutions are obtained. The obtained results provide partial answers to some open questions posed by Gal and Varma (2020), as well as Luchko and Yamamoto (2016). Furthermore, possible quasi-linear extensions of the obtained results are discussed, and some open questions are presented.

Theorems & Definitions (8)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• proof : Proof of Theorem \ref{['Th2']}
• proof : Proof of Theorem \ref{['Th1']}