Nonlocal final value problem governed by semilinear anomalous diffusion equations

Abstract

Our goal is to establish some sufficient conditions for the solvability of the nonlocal final value problem involving a class of partial differential equations, which describes the anomalous diffusion phenomenon. Our analysis is based on the theory of completely positive functions, resolvent operators and fixed point arguments in suitable function spaces. Especially, utilizing the regularity of resolvent operators, we are able to deal with non-Lipschitz cases. The obtained results, in particular, extend recent ones proved for fractional diffusion equations.

Source identification problems for abstract semilinear nonlocal differential equations

“…The forward problem with nonlocal differential version gives us the models appearing in mathematical physics in the context of anomalous diffusion ones, fluid flow through porous media; see, e.g. [1,10,11,16,29,21,23,26]. We would like also to emphasize that, in constract to the local differential model, which the derivative can be determined by observations in the neighboorhoods, the nonlocal differential equations can be obtained from the observations in the whole of the interests.…”

“…In the following, we collect some important properties of s(•, µ) and r(•, µ) which were given in [29].…”