Long-time dynamics for time-nonlocal generalized Rayleigh-Stokes equations

Abstract

In this paper, we consider an autonomous semi-dynamical system driven by semilinear time-nonlocal evolution equations, these type equations are used to describe the Rayleigh-Stokes problem for a non-Newtonain fluid to a generalized second grade fluid. We first investigate the global well-posedness of solutions consisting of global Lipschitz condition by a weighted space $\mathcal{C}$. Utilizing the topology convergence on compact subsets of $\mathcal{C}$, we construct a semi-dynamical system that satisfies the semi-group structure. It also is shown that this semi-dynamical system has an attracting set when the vector field function satisfies a dissipativity condition and a local Lipschitz condition. With the asymptotic compactness, we also establish the existence of generalized attractors in ${\mathcal{C}}_{\hat{I}}\pm$ of subspace of $\mathcal{C}$ the weighted norm.

42 pages