Reconstruction of a bounded variation convolution kernel in an abstract wave equation

摘要

We consider an abstract linear integrodifferential hyperbolic problem, with an unknown scalar convolution kernel. This lack of information is compensated by the knowledge, for any time, of a certain expression, depending on the solution u and on the kernel k. Under suitable assumptions, we prove a result of global existence and uniqueness of the pair (u, k). k is determined in the class of bounded variation (not necessarily continuous) functions. The abstract results are applied to some concrete hyperbolic systems, and in these concrete cases, the aforementioned expression is a certain flux through the boundary.