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

Abstract

We consider the second order Cauchy problem $$u''+\m{u}Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty )\to [0,+\infty )$ 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 $u_0$ and $u_1$ 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 $<A{u}_0,u_1>\ne 0$, or $|A^{1/2}u_{1}|^{2}\neq\m{u_{0}}|Au_{0}|^{2}$, then the local solution is unique even if $m$ is not Lipschitz continuous.

15 pages