An example of non-uniqueness for a hyperbolic equation with non-Lipschitz-continuous coefficients. (Q1411709)

A second order strictly hyperbolic linear partial differential operator \(P\) in the \(n+1\) dimensional Euclidean space is considered. If the coefficients in the principal part of the operator satisfy the Lipschitz condition, then P has the uniqueness in the Cauchy problem. In [\textit{F. Colombini}, \textit{E. Jannelli} and \textit{S. Spagnolo}, Non-uniqueness in hyperbolic Cauchy problems, Ann. Math. (2) 126, 495--524 (1987; Zbl 0649.35051)], one shows the existence of hyperbolic operators with Hölder continuous coefficients in the principal part that does not have the uniqueness property. In the present article one tries to improve the above result by better pointing out the connection between the regularity of the coefficients in the principal part of the operator and the uniqueness in the Cauchy problem.