Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation

Abstract

We consider the recovery of a potential associated with a semi-linear wave equation on Rn+1, n≥1. We show that an unknown potential a(x,t) of the wave equation □u+aum=0 can be recovered in a Hölder stable way from the map u|∂Ω×[0,T]↦〈ψ,∂νu|∂Ω×[0,T]〉L2(∂Ω×[0,T]). This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function ψ. We also prove similar stability result for the recovery of a when there is noise added to the boundary data. The method we use is constructive and it is based on the higher order linearization. As a consequence, we also get a uniqueness result. We also give a detailed presentation of the forward problem for the equation □u+aum=0.