A final value problem with a non-local and a source term: regularization by truncation

Abstract: This paper is concerned with recovering the solution of a final value problem associated with a parabolic equation involving a non linear source and a non-local term, which to the best of our knowledge has not been studied earlier. It is shown that the considered problem is ill-posed, and thus, some regularization method has to be employed in order to obtain stable approximations. In this regard, we obtain regularized approximations by solving some non linear integral equations which is derived by considering a truncated version of the Fourier expansion of the sought solution. Under different Gevrey smoothness assumptions on the exact solution, we provide parameter choice strategies and obtain the error estimates. A key tool in deriving such estimates is a version of Gr{\"o}nwalls' inequality for iterated integrals, which perhaps, is proposed and analysed for the first time.