Sharp extensions and algebraic properties for solution families of vector-valued differential equations

Abstract

In this paper we show the unexpected property that extension from local to global without loss of regularity holds for the solutions of a wide class of vector-valued differential equations, in particular for the class of fractional abstract Cauchy problems in the subdiffusive case. The main technique is the use of the algebraic structure of these solutions, which are defined by new versions of functional equations defining solution families of bounded operators. The convolution product and the double Laplace transform for functions of two variables are useful tools which we apply also to extend these solutions. Finally we illustrate our results with different concrete examples.

Fractional-Order Susceptible-Infected Model: Definition and Applications to the Study of COVID-19 Main Protease

“…Also, it is well-known [9,18,34] (or more recently [2,3,5]) that the revious Mittag-Leffler matrix functions satisfy…”

Fractional Cauchy problem with memory effects

“…See also Li and Peng [23] where this approach has been taken in the context of resolvent families for fractional Cauchy problems without memory terms. In a general context, both operator families, the resolvent and integral resolvent, have been studied in different papers, see [1, 2, 23, 24 26, 29] for more information.…”

Uniform stability for fractional Cauchy problems and applications

“…These types of operator families are framed into the theory of solutions of abstract Volterra integral equations studied in the book of J. Prüss [43]. Also, algebraic, extension and subordination properties of them have been studied in [2,3,6,7,13,28,29,30,31,35].…”

“…See also Li and Peng [30]. In a general context, both the resolvent and integral resolvent have been studied in di↵erent papers ( [1,2,30,31,35,43]).…”