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

We consider the second order Cauchy problemu″+m(|A1/2u|2)Au=0,u(0)=u0,u′(0)=u1, where m:[0,+∞)→[0,+∞) is a continuous function, and A is a self-adjoint nonnegative operator with dense domain on a Hilbert space.It is well known that this problem admits local-in-time solutions provided that u0u0 and u1u1 are regular enough, depending on the continuity modulus of m. It is also well known that the solution is unique when m is locally Lipschitz continuous.In this paper we prove that if either 〈Au0,u1〉≠0〈Au0,u1〉≠0, or |A1/2u1|2≠m(|A1/2u0|2)|Au0|2|A1/2u1|2≠m(|A1/2u0|2)|Au0|2, then the local solution is unique even if m is not Lipschitz continuous.