Fractional approximations of abstract semilinear parabolic problems.

In this paper we study the abstract semilinear parabolic problem of the form du/dt + Au = ƒ(u), as the limit of the corresponding fractional approximations du/dt + A^αu = ƒ(u), in a Banach space X, where the operator A : D(A) ⊂ X → X is a sectorial operator in the sense of Henry [22]. Under suitable assumptions on nonlinearities ƒ : X^α → X (X^α : = D(A^α)), we prove the continuity with rate (with respect to the parameter α) 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. [ABSTRACT FROM AUTHOR]