Reconstruction of an unknown source parameter in a semilinear parabolic problem

In this paper, a semilinear parabolic problem with an unknown time-dependent source function p(t) is studied. This missing parameter is reconstructed from a given measurement of the total energy/mass in the domain. The existence and uniqueness of a solution in suitable function spaces is established under minimal regularity assumptions on the data. A numerical time-discrete scheme to approximate the unique weak solution and the unknown source parameter is designed and convergence of the approximations is proved. Finally, the theoretically obtained results are supported by a numerical experiment.