An identification problem governed by nonlinear fractional mobile-immobile equation, Part I: Solvability

In this study, we address the inverse source problem of identifying a space-dependent parameter in nonlinear fractional mobile-immobile (FrM-IM) equations. The inverse problem is resolved using supplementary measurements taken at the final time, which are permitted to depend implicitly on the system’s state. This work is presented in two parts. In Part I, we first establish regularity estimates for resolvent operators associated with the linear FrM-IM equation under Dirichlet boundary conditions. Due to these estimates, we employ fixed-point arguments and local analysis on Hilbert scales to rigorously prove the existence and uniqueness of solutions to the nonlinear inverse problem. In Part II (to be addressed separately), under sufficient regularity assumptions on the final datum and the governing nonlinearities, we demonstrate that the solution derived in Part I is, in fact, a strong solution. Our analysis advances the theoretical framework for FrM-IM equations by unifying resolvent operator theory with nonlinear fixed-point methods, thereby providing a foundation for addressing inverse problems in nonlocal transport phenomena.