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

Abstract

We consider the recovery of a potential associated with a semi-linear wave equation on ${\mathbb{R}}^{n+1}$, $n\ge 1$. We show a Hölder stability estimate for the recovery of an unknown potential $a$ of the wave equation $\square u+a{u}^m=0$ from its Dirichlet-to-Neumann map. We show that an unknown potential $a(x,t)$, supported in $\Omega \times [t_1,t_2]$, of the wave equation $\square u+a{u}^m=0$ can be recovered in a Hölder stable way from the map $u{|}_{\partial \Omega \times [0,T]}\mapsto ⟨\psi ,{\partial }_{\nu }u{|}_{\partial \Omega \times [0,T]}{⟩}_{L^2(\partial \Omega \times [0,T])}$. This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function $\psi$. 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 $\square u+a{u}^m=0$.

arXiv comment32 pages

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