Uniqueness for inverse source problems for fractional diffusion-wave equations by data during not acting time

TL;DR

This paper proves the uniqueness of determining a spatial source in fractional diffusion-wave equations using data collected during periods when the source is inactive, based on decay rates as time approaches infinity.

Contribution

It establishes the first general uniqueness result for inverse source problems with data during non-acting times for fractional diffusion-wave equations.

Findings

Uniqueness holds if data decay faster than 1/t^p for any p in natural numbers.

The proof utilizes analyticity and asymptotic analysis of solution-generated functions.

Data collected during source inactivity can determine the spatial source uniquely.

Abstract

We consider fractional diffusion-wave equations with source term which is represented in a form of a product of a temporal function and a spatial function. We prove the uniqueness for inveres source problem of determining spatially varying factor by decay of data as the time tends to $\infty$, provided that the source does not work during the observations. Our main result asserts the uniqueness if data decay more rapidly than $\left(\frac{1}{t^p}\right)$ with any $p\in \N$ as $t\to\infty$. Date taken not from the initial time are realistic but the uniqueness was not known in general. The proof is based on the analyticity and the asymptotic behavior of a function generated by the solution.