$C^\infty$ well-posedness of higher order hyperbolic pseudo-differential equations with multiplicities

TL;DR

This paper establishes conditions for the smooth well-posedness of higher order hyperbolic pseudo-differential equations with variable multiplicities, extending the theory to arbitrary dimensions and non-diagonalisable systems.

Contribution

It provides new sufficient conditions (Levi conditions) ensuring $C^ abla$ well-posedness for complex hyperbolic systems with multiplicities, using advanced Fourier integral operator techniques.

Findings

Identified Levi conditions for well-posedness

Extended results to arbitrary space dimensions

Compared with existing literature on second and third order equations

Abstract

In this paper, we study higher order hyperbolic pseudo-differential equations with variable multiplicities. We work in arbitrary space dimension and we assume that the principal part is time-dependent only. We identify sufficient conditions on the roots and the lower order terms (Levi conditions) under which the corresponding Cauchy problem is $C^\infty$ well-posed. This is achieved via transformation into a first order system, reduction into upper-triangular form and application of suitable Fourier integral operator methods previously developed for hyperbolic non-diagonalisable systems. We also discuss how our result compares with the literature on second and third order hyperbolic equations.