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

TL;DR

This paper proves a uniqueness result for solutions to Kirchhoff equations with non-Lipschitz nonlinear terms, showing uniqueness under certain initial conditions even without Lipschitz continuity of the nonlinear function.

Contribution

It establishes new uniqueness conditions for Kirchhoff equations with non-Lipschitz nonlinearities, extending known results beyond Lipschitz cases.

Findings

Uniqueness holds if specific initial conditions are met.

Local solutions are unique even when the nonlinear term is not Lipschitz.

Provides conditions for uniqueness based on initial data properties.

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 $<Au_{0},u_{1}>\neq 0$, 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.