FRACTIONAL APPROXIMATIONS OF ABSTRACT SEMILINEAR PARABOLIC PROBLEMS

In this paper we study the abstract semilinear parabolic problem of the form du/dt + Au = f(u), as the limit of the corresponding fractional approximations du/dt + A(alpha)u = f(u), in a Banach space X, where the operator A : D(A) subset of X -> X is a sectorial operator in the sense of Henry [22]. Under suitable assumptions on nonlinearities f : X-alpha -> 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.