Final value problem for fractional reaction-subdiffusion equations

We investigate the existence of a mild solution to the final value problem for a class of fractional reaction-subdiffusion nonlinear equations, where the nonlinearity may take weak values. We want to demonstrate the unique existence of a mild solution by using the Banach fixed-point theorem. In order to do this, we construct some new estimates for the resolvent function and the resolvent operator, based on the existing resolvent theory. From our point of view, the nonlinearity, which takes values in Hilbert scales, presents some technical difficulties but allows us to examine broader classes of problems, since by which it can contain a polynomial or gradient term arising from various physical circumstances.