Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data

TL;DR

This paper addresses the inverse problem of uniquely recovering time-dependent damping coefficients and potentials in wave equations using partial boundary data, providing theoretical guarantees for the uniqueness of such reconstructions.

Contribution

It establishes the global unique determination of a broad class of time-dependent damping coefficients and potentials from partial boundary observations in wave equations.

Findings

Unique determination of damping coefficient a(t,x) in W^{1,p}(Q), p>n+1

Unique determination of potential q(t,x) in L^ fty(Q)

Results applicable to partial boundary data scenarios

Abstract

We consider the inverse problem of determining a time-dependent damping coefficient $a$ and a time-dependent potential $q$, appearing in the wave equation $\partial_t^2u-\Delta_x u+a(t,x)\partial_tu+q(t,x)u=0$ in $Q=(0,T)\times\Omega$, with $T>0$ and $\Omega$ a $ \mathcal C^2$ bounded domain of $\mathbb R^n$, $n\geq2$, from partial observations of the solutions on $\partial Q$. More precisely, we look for observations on $\partial Q$ that allow to determine uniquely a large class of time-dependent damping coefficients $a$ and time-dependent potentials $q$ without involving an important set of data. We prove global unique determination of $a\in W^{1,p}(Q)$, with $p>n+1$, and $q\in L^\infty(Q)$ from partial observations on $\partial Q$.