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

Abstract

We consider the second order Cauchy problem u ″ + m ( | A 1 / 2 u | 2 ) A u = 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 , 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 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 〉 ≠ 0 , or | A 1 / 2 u 1 | 2 ≠ m ( | A 1 / 2 u 0 | 2 ) | A u 0 | 2 , then the local solution is unique even if m is not Lipschitz continuous.