Heat and wave type equations with non-local operators, I. Compact Lie groups

TL;DR

This paper establishes existence, uniqueness, and analytical solutions for heat and wave equations involving non-local operators on compact Lie groups, including asymptotic behavior and examples.

Contribution

It introduces a framework for solving heat and wave equations with non-local operators on compact Lie groups, extending classical results to more general operators.

Findings

Solutions exist and are unique under specified conditions.

Asymptotic estimates for large-time behavior are provided.

Solutions are characterized in appropriate function spaces.

Abstract

We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty$. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat type equation as well.