Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\mathbb{R}^d$

TL;DR

This paper establishes optimal decay estimates for solutions to non-local in time subdiffusion equations, including the time-fractional case, revealing a critical dimension phenomenon and employing multiple analytical techniques.

Contribution

It provides the first comprehensive decay estimates for a broad class of non-local in time subdiffusion equations, highlighting the critical dimension effect and applying diverse analytical methods.

Findings

Decay rates differ from classical heat equations.

Existence of a critical dimension phenomenon.

Applicability to various kernels, including ultraslow diffusion.

Abstract

We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in $\mathbb{R}^d$. An important special case is the time-fractional diffusion equation, which has seen much interest during the last years, mostly due to its applications in the modeling of anomalous diffusion processes. We follow three different approaches and techniques to study this particular case: (A) estimates based on the fundamental solution and Young's inequality, (B) Fourier multiplier methods, and (C) the energy method. It turns out that the decay behaviour is markedly different from the heat equation case, in particular there occurs a {\em critical dimension phenomenon}. The general subdiffusion case is treated by method (B) and relies on a careful estimation of the underlying relaxation function. Several examples of kernels, including the ultraslow diffusion case, illustrate our results.