Unique determination of a time-dependent potential for wave equations from partial data

TL;DR

This paper proves that a time-dependent potential in a wave equation can be uniquely identified using partial boundary observations, advancing inverse problem theory.

Contribution

It establishes the global uniqueness of determining a time-dependent potential in a wave equation from partial boundary data.

Findings

Unique determination of the potential $q$ from partial boundary observations.

The potential $q$ can be recovered globally in the domain.

The result applies to bounded domains with smooth boundaries.

Abstract

We consider the inverse problem of determining a time-dependent potential $q$, appearing in the wave equation $\partial_t^2u-\Delta u+q(t,x)u=0$ in $Q=(0,T)\times\Omega$ with $\Omega$ a $C^2$ bounded domain of $\mathbb R^n$, $n\geq2$, from partial observations of the solutions on $\partial Q$. We prove global unique determination of a coefficient $q\in L^\infty(Q)$ from these observations.