Evolutionary integro-differential equations of scalar type on locally compact groups

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of solutions to scalar evolutionary integro-differential equations on locally compact groups, including heat and wave equations, with applications to nonlinear PDEs.

Contribution

It extends the analysis of scalar evolutionary equations to locally compact groups with general operators, providing new estimates and well-posedness results for nonlinear equations.

Findings

Established existence and uniqueness of solutions.

Derived sharp norm estimates and asymptotic behavior.

Provided Strichartz-type estimates for heat and wave equations.

Abstract

We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact separable unimodular group $G$ governed by any positive left invariant operator (unbounded and either with discrete or continuous spectrum) on $G$. We complement our studies by proving some time-space (Strichartz type) estimates for the classical heat equation, and its time-fractional counterpart, as well as for the time-fractional wave equation. The latter estimates allow us to give some results about the well-posedness of nonlinear partial integro-differential equations. We provide many examples of the results by considering particular equations, operators and groups through the whole paper.