Some direct and inverse source problems in nonlinear evolutionary PDEs with Volterra operators

Abstract

This paper deals with direct and inverse source problems for parabolic or byperbolic PDEs containing nonlinear Volterra operators (including the variable order time-fractional derivatives). We show the well posedness (existence and uniqueness of a solution) of various settings. The proof-technique is based on the contraction mapping theorem and on a bootstrap argument, which ensures the existence of a global in time solution.