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

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 H & ouml;lder stable way from the map u|onnx[0,T] ???-> (11, avu|ac >= x[0,T])L2(oc >= x[0,T]). This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function ???. 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. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Superscript/Subscript AvailablePeer reviewe