A uniqueness result for Kirchhoff equations with non-Lipschitz nonlinear term

We consider the second order Cauchy problem for the Kirchhoff equation. It is well known that this problem admits local-in-time solutions provided that the initial data are regular enough, depending on the continuity modulus of the nonlinear term m. It is also well known that the solution is unique when m is locally Lipschitz continuous. In this paper we prove that if the initial data verify some algebraic condition, then the local solution is unique even if m is not Lipschitz continuous.