Asymptotic expansion of solutions to an inverse problem of parabolic type

Abstract

We consider an abstract linear parabolic problem, with a scalar part $f$ of the source term which is unknown. This lack of information is compensated by the knowledge, for any time, of the value of a certain functional $\Phi$ when applied to the solution. Under suitable assumptions, we prove a result of global existence and uniqueness of the solution $(u,f)$. Moreover, if the coefficients of the system admit an asymptotic expansion, the same holds for $(u,f)$. The abstract results are applied to general parabolic mixed Cauchy-boundary value problems.