On global solvability and regularity for generalized Rayleigh-Stokes equations with history-dependent nonlinearities

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to generalized Rayleigh-Stokes equations with history-dependent nonlinearities, employing fixed point methods and fractional Sobolev space embeddings, and applies findings to an inverse source problem.

Contribution

It introduces new results on solvability and regularity for equations with history-dependent nonlinearities in Hilbert scales, expanding understanding of such complex PDEs.

Findings

Proved existence and Hölder regularity of solutions.

Established fixed point framework for nonlinear history-dependent PDEs.

Applied results to an inverse source problem.

Abstract

We are concerned with the initial value problem governed by generalized Rayleigh-Stokes equations, where the nonlinearity depends on history states and takes values in Hilbert scales of negative order. The solvability and H\"older regularity of solutions are proved by using fixed point arguments and embeddings of fractional Sobolev spaces. An application to a related inverse source problem is given.