Sharp extensions for convoluted solutions of wave equations

TL;DR

This paper develops sharp extensions for convoluted solutions of wave equations in Banach spaces using algebraic structures and Duhamel's formula, with applications to abstract wave equations.

Contribution

It introduces new algebraic extension techniques for convoluted wave solutions in Banach spaces, enhancing understanding of their structure and applications.

Findings

Established sharp extensions for convoluted wave solutions

Defined algebra homomorphisms for convolution products

Applied results to concrete abstract wave equations

Abstract

In this paper we give sharp extensions of convoluted solutions of wave equations in abstract Banach spaces. The main technique is to use the algebraic structure, for convolution products $\ast$ and $\ast_c$, of these solutions which are defined by a version of the Duhamel's formula. We define algebra homomorphisms, for the convolution product $\ast_c$, from a certain set of test-functions and apply our results to concrete examples of abstract wave equations.