On Hyperbolic Equations with Space-Dependent Coefficients: \(C^\infty \) Well-Posedness and Levi Conditions

This note 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 the 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 conditions in Oleinik (Commun Pure Appl Math 23, 569–586, 1970) to orders higher than 2.