Well-posedness for a class of pseudo-differential hyperbolic equations on the torus

TL;DR

This paper proves the well-posedness of a class of pseudo-differential hyperbolic equations on the torus, including fractional Laplacians, using toroidal pseudo-differential calculus to establish regularity, existence, and uniqueness.

Contribution

It introduces a framework for analyzing hyperbolic equations with fractional Laplacians on the torus, extending previous results to a broader class of pseudo-differential operators.

Findings

Established well-posedness in Sobolev spaces

Derived regularity estimates for solutions

Proved existence and uniqueness of solutions

Abstract

In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus