We study the long time behavior of solutions for time-fractional pseudo-parabolic equation involving time-varying delays and nonlinear pertubations, where the nonlinear term allows to have a superlinear growth. Concerning the associated linear problem, we establish a variation of parameters formula of mild solution and prove some regularity estimates of resolvent operators. In addition, thanks to local estimates on Hilbert scales, fixed point arguments and a new Halanay type inequality, we obtain some results on the global solvability, stability, dissipativity and the existence of decay solutions to our problem.

Our aim in this paper is to give a sufficient condition ensuring the finite-time attractivity for the zero solution to semilinear functional differential inclusions in Banach spaces, in the case where the nonlinearity function possibly has superlinear growth. Our analysis is based on the semigroup theory, the fixed point principle for condensing multi-valued maps, and local estimates of solutions. The abstract results will be applied to a class of polytope inclusions in $C_0$ setting.