On stability for semilinear generalized Rayleigh-Stokes equation involving delays

We consider a functional semilinear Rayleigh-Stokes equation involving fractional derivative. Our aim is to analyze some circumstances, in those the global solvability, and asymptotic behavior of solutions are addressed. By establishing a 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.