The Cauchy Problem for properly hyperbolic equations in one space variable

TL;DR

This paper investigates the well-posedness of the Cauchy problem for higher order weakly hyperbolic equations with principal symbols depending on one spatial variable, establishing conditions for smooth solutions.

Contribution

It provides new well-posedness results for a class of weakly hyperbolic equations under specific Levi conditions and characteristic root inequalities.

Findings

Well-posedness in -infinity for the considered equations

Conditions on characteristic roots ensuring stability

Extension of classical results to higher order weakly hyperbolic equations

Abstract

In this paper we consider the Cauchy problem for higher order weakly hyperbolic equations. We assume that the principal symbol depends only on one space variable and the characteristic roots $\tau_j$ verify the inequality \[\tau_j^2(x) + \tau_k^2(x) \le M \bigl(\tau_j(x)-\tau_k(x)\bigr)^2\] for some constant $M$ independent of $x$. We prove that if the lower order terms verify a suitable Levi condition, the Cauchy problem is well-posed in $\mathcal{C}$-infinity.