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

We consider the recovery of a potential associated with a semi-linear wave equation on $\mathbb{R}^{n+1}$, $n\geq 1$. We show a H\"older 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 H\"older stable way from the map $u|_{\partial \Omega\times [0,T]}\mapsto \langle\psi,\partial_\nu u|_{\partial \Omega\times [0,T]}\rangle_{L^2(\partial \Omega\times [0,T])}$. 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 $\square u +a u^m=0$.