Maximal time regularity for degenerate evolution integro-differential equations

Abstract

We provide maximal time regularity properties for the solutions to a class of degenerate firstorder integro-differential Cauchy problems in a Banach space X . In particular, we show that an additional condition of space regularity on the data it suffices for restoring the loss of time regularity which arises naturally when dealing with the degenerate case.

On the Behaviour of Singular Semigroups in Intermediate and Interpolation Spaces and Its Applications to Maximal Regularity for Degenerate Integro-Differential Evolution Equations

“…Regrettably (cf. the appendix below), we have found some inaccuracies in [20,Section 4], and for this reason we must indicate some changes to that paper. On the other side, fortunately, the basic idea in [20] is correct and remedy can be applied to all the inappropriate items.…”

“…Such a theory has been then successfully applied to questions of maximal regularity for the solutions to a wide class of degenerate integro-and non-integrodifferential equations. We quote [2,[19][20][21][22][23] where, in general and unless = 1, it is shown that the time regularity of the solutions decreases with respect to that of the data. In this respect, we mention the recent results in [20] where, under an additional condition of space regularity on the data and provided that and are large enough, the loss of time regularity is restored.…”

Hölder continuous solutions for Sobolev type differential equations

“…Existence and uniqueness of Hölder continuous solutions for equations in the form of have been extensively studied in the literature. See –, and the references therein. The obtained results give sufficient conditions for the existence and uniqueness of Hölder solutions to Equation , but leave as an open problem to characterize the well‐posedness (or maximal regularity) in terms of hypothesis in the modified resolvent operator of the operators M and A .…”

“…In some previous works, to establish the existence and uniqueness of solutions to Equation some assumptions on operators A and M are considered: and A admits a continuous inverse operator , , and M has the bounded inverse , and M has the compact inverse , . …”

“…(i) D(A) ⊆ D(M) and A admits a continuous inverse operator A −1 [11], [12], (ii) D(A) ⊆ D(M) and M has the bounded inverse [14], (iii) D(M) ⊆ D(A) and M has the compact inverse [4], [5].…”