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

Abstract

We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation ${\partial }_t^{2}u-{\Delta }_{x}u+q(t,x)u=0$ in $Q=(0,T)\times \Omega$ with $T>0$ and $\Omega$ a ${\mathcal{C}}^2$ bounded domain of ${\mathbb{R}}^n$, $n\ge 2$. 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.

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