The inverse problem for a wave equation with a power law nonlinearity and a potential depending on spatial and temporary variables

The inverse problem of determining the coefficient $q(x,t)$ of the wave equation $\partial^2_t u-\partial^2_x u-q(x,t)u^{\gamma+1}=0$ is considered when $(x,t)\in \mathbb{R}^{+}\times (0,T]$, $\gamma\ge0$. The properties of a solution of a direct problem are studied and an existence and uniqueness theorem is proved. For the inverse problem a local existence theorem is stated and a global stability estimates is found.

УДК 517.958

Keywords: Inverse problem, local existence, global stability, power nonlinearity