On higher order hyperbolic equations with space-dependent coefficients: $C^\infty$ well-posedness and Levi conditions

TL;DR

This paper establishes Sobolev well-posedness for higher order hyperbolic equations with space-dependent coefficients, extending Levi conditions to higher orders and addressing multiplicities in multiple dimensions.

Contribution

It generalizes Olienik's Levi conditions to higher order hyperbolic equations with space-dependent coefficients, proving well-posedness in Sobolev spaces.

Findings

Sobolev well-posedness under Levi conditions

Extension of Levi conditions to higher orders

Handling of multiplicities in hyperbolic equations

Abstract

This paper contributes to the wider study of hyperbolic equations with multiplicities. We focus here on some classes of higher order hyperbolic equations with space dependent coefficients in any space dimension. We prove Sobolev well-posedness of the corresponding Cauchy problem (with loss of derivatives due to the multiplicities) under suitable Levi conditions on the lower order terms. These conditions generalise the well known Olienik's conditions in \cite{O70} to orders higher than $2$.