Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods

TL;DR

This paper establishes sharp decay estimates for solutions to non-local subdiffusion equations, including fractional and ultraslow diffusion, using energy methods and a novel inequality for integro-differential operators.

Contribution

It introduces a new inequality for integro-differential operators and applies energy methods to derive optimal decay estimates for a broad class of non-local subdiffusion equations.

Findings

Decay rates differ significantly from classical parabolic equations.

Results apply to time-fractional and ultraslow diffusion models.

Method extends to certain quasilinear equations like the p-Laplace and porous medium equations.

Abstract

We prove sharp estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the time-fractional and ultraslow diffusion equation, which have seen much interest during the last years, mostly due to their applications in the modeling of anomalous diffusion. We study the case where the equation is in divergence form with bounded measurable coefficients. Our proofs rely on energy estimates and make use of a new and powerful inequality for integro-differential operators of the form $\partial_t (k\ast \cdot)$. The results can be generalized to certain quasilinear equations. We illustrate this by looking at the time-fractional $p$-Laplace and porous medium equation. Here it turns out that the decay behaviour is markedly different from that in the classical parabolic case.