# Determination of singular time-dependent coefficients for wave equations from full and partial data

## TL;DR

This paper proves the unique determination of unbounded, time-dependent singular coefficients in wave equations from boundary data, advancing inverse problem theory for nonlinear wave equations.

## Contribution

It is the first to establish unique recovery of unbounded, singular, time-dependent coefficients in wave equations from boundary observations.

## Key findings

- Unique determination of unbounded coefficients from boundary data
- Reduction of data requirements for coefficient recovery
- First result on unbounded time-dependent coefficients in wave equations

## Abstract

We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation $\partial_t^2u-\Delta_x u+q(t,x)u=0$ in $Q=(0,T)\times\Omega$ with $T>0$ and $\Omega$ a $ \mathcal C^2$ bounded domain of $\mathbb R^n$, $n\geq2$. We start by considering the unique determination of some singular time-dependent coefficients from observations on $\partial Q$. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.