The Super-Diffusive Singular Perturbation Problem

In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem (Pε)εαDε t uε(t) + u ε(t) = Auε(t), t ε [0, T], 1 < α < 2, ε > 0, for the parabolic equation (P)u 0(t) = Au0(t), t ε [0, T], in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem (Pε) has a unique solution uεe(t) for each small ε > 0. Moreover uε(t) converges to uε(t) as ε rarr; 0+, the unique solution of equation (P).. © 2020 by the authors.