On stability for semilinear generalized Rayleigh-Stokes equation involving delays

TL;DR

This paper investigates the stability and long-term behavior of solutions to a fractional semilinear Rayleigh-Stokes equation with delays, establishing conditions for stability, dissipativity, and decay solutions.

Contribution

It introduces a new Halanay type inequality and applies fixed point methods to analyze stability and existence of decay solutions for the equation.

Findings

Solutions are globally stable under certain conditions.

A new inequality helps establish dissipativity.

Existence of a compact set of decay solutions is proven.

Abstract

We consider a functional semilinear Rayleigh-Stokes equation involving fractional derivative. Our aim is to analyze some circumstances, in those the global solvability and some results on asymptotic behavior of solutions take place. By establishing a new Halanay type inequality, we show the dissipativity and asymptotic stability of solutions to our problem. In addition, we prove the existence of a compact set of decay solutions by using local estimates and fixed point arguments.