Global well-posedness of the Kirchhoff equation and Kirchhoff systems

TL;DR

This paper reviews the current state of research on the global well-posedness of Kirchhoff equations and systems, including small and large data cases, boundary value problems, and open challenges in the field.

Contribution

It provides a comprehensive overview of known results and open problems related to the global well-posedness of Kirchhoff equations and systems.

Findings

Review of global well-posedness results for small data

Discussion of initial-boundary value problems in exterior domains

Summary of open problems for large data cases

Abstract

This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.