Decay rates and initial values for time-fractional diffusion-wave equations

TL;DR

This paper analyzes the decay rates of solutions to time-fractional diffusion-wave equations, establishing bounds and characterizations of initial values based on asymptotic behaviors for different fractional orders.

Contribution

It provides new bounds on solution decay rates for fractional orders and characterizes initial values based on these decay behaviors, extending classical diffusion results.

Findings

Decay rate bounds for different fractional orders.

Characterization of initial values from decay rates.

Use of eigenfunction expansions and Mittag-Leffler asymptotics.

Abstract

We consider a solution $u(\cdot,t)$ to an initial boundary value problem for time-fractional diffusion-wave equation with the order $\alpha \in (0,2) \setminus \{ 1\}$ where $t$ is a time variable. We first prove that a suitable norm of $u(\cdot,t)$ is bounded by $\frac{1}{t^{\alpha}}$ for $0<\alpha<1$ and $\frac{1}{t^{\alpha-1}}$ for $1<\alpha<2$ for all large $t>0$. Moreover we characterize initial values in the cases where the decay rates are faster than the above critical exponents. Differently from the classical diffusion equation $\alpha=1$, the decay rate can give some local characterization of initial values. The proof is based on the eigenfunction expansions of solutions and the asymptotic expansions of the Mittag-Leffler functions for large time.