# C∞ Well-Posedness of Higher Order Hyperbolic Pseudo-Differential Equations with Multiplicities

## TL;DR

This paper investigates the well-posedness of higher order hyperbolic pseudo-differential equations with multiplicities in arbitrary space dimensions.

## Contribution

The study establishes sufficient conditions for $C^\infty$ well-posedness of the Cauchy problem using transformation and Fourier integral operator methods.

## Key findings

- Sufficient conditions (Levi conditions) for $C^\infty$ well-posedness are identified.
- A method involving transformation into a first order system and upper-triangular reduction is applied successfully.
- The results are compared with existing literature on second and third order hyperbolic 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 \documentclass[12pt]{minimal}
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\begin{document}$C^{\infty }$\end{document}C∞ 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.