A parabolic inverse source problem with a dynamical boundary condition

Abstract

An inverse source problem for the heat equation is studied in a bounded domain. A dynamical nonlinear boundary condition (containing the time derivative of a solution) is prescribed on one part of the boundary. This models a non-perfect contact on the boundary. The missing purely time-dependent source is recovered from an additional integral measurement. The global in time existence and uniqueness of a solution in corresponding function spaces is addressed using the backward Euler method for the time discretization. Error estimates for time-discrete approximations are derived.