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An inverse problem for the strongly damped wave equation with memory

We consider the identification problem which consists of determining simultaneously the function u and the convolution kernel h in the strongly damped wave equation with memory given suitable initial boundary conditions and additional measurements of u represented by where Ω is an open bounded set in and f, G, ϕ and g are given functions. We prove a global-in-time existence and unique result in the case where f(u, ∇u) has a suitable regularity and sublinear growth using a technique recently developed for parabolic equations. In the case that we assume only regularity conditions on f(u, ∇u), but we do not give any growth condition, we are able to prove a local-in-time existence theorem and a global-in-time uniqueness result.