Fractional approximations of abstract semilinear parabolic problems
Flank D. M. Bezerra, Alexandre N. Carvalho, Marcelo J. D. Nascimento
Abstract
In this paper we study the abstract semilinear parabolic problem of the form $ \frac{du}{dt}+Au = f(u), $ as the limit of the corresponding fractional approximations $ \frac{du}{dt} + A^{\alpha}u = f(u), $ in a Banach space $ X $, where the operator $ A:D(A) \subset X \to X $ is a sectorial operator in the sense of Henry [22]. Under suitable assumptions on nonlinearities $ f:X^\alpha\to X $ ($ X^\alpha: = D(A^\alpha $)), we prove the continuity with rate (with respect to the parameter $ \alpha $) for the global attractors (as seen in Babin and Vishik [4] Chapter 8, Theorem 2.1). As an application of our analysis we consider a fractional approximation of the strongly damped wave equations and we study the convergence with rate of solutions of such approximations.