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

TL;DR

This paper demonstrates that solutions to a broad class of vector-valued differential equations, including fractional Cauchy problems, can be extended from local to global solutions without losing regularity, using algebraic and transform techniques.

Contribution

It introduces new algebraic methods and functional equations to extend solutions globally for vector-valued differential equations, especially fractional cases.

Findings

Extension from local to global solutions without regularity loss

Use of convolution product and double Laplace transform for solution extension

Application of methods to concrete examples

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.