Asymptotic behavior of solutions to space-time fractional diffusion equations

TL;DR

This paper analyzes the long-time decay behavior of solutions to space-time fractional diffusion equations, showing that the decay rate is primarily determined by the order of the fractional derivative, with implications for both unbounded and bounded domains.

Contribution

The paper provides a rigorous analysis of the asymptotic decay rates of solutions to space-time fractional diffusion equations using Laplace transform techniques, extending understanding to bounded domains.

Findings

Decay rate is dominated by the order of the time-fractional derivative.

Solutions exhibit specific asymptotic decay behavior as time approaches infinity.

Decay behavior is characterized in both unbounded and bounded spatial domains.

Abstract

This article discusses the analyticity and the long-time asymptotic behavior of solutions to space-time fractional diffusion equations in $\mathbb{R}^d$. By a Laplace transform argument, we prove that the decay rate of the solution as $t\to\infty$ is dominated by the order of the time-fractional derivative. We consider the decay rate also in a bounded domain.