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

作者：Ghisi Marina; Gobbino Massimo^*

来源：Advances in Mathematics, 2010, 223(4): 1299-1315.

DOI：10.1016/j.aim.2009.09.017

• Uniqueness
• Integro-differential hyperbolic equation
• Continuity modulus
• Kirchhoff equations
• Gevrey spaces

We consider the second order Cauchy problem
u '' + m(vertical bar A(1/2)u vertical bar(2))Au = 0, u(0) = u(0), u'(0) = u(1),
where m: [0, +infinity) -> [0, +infinity) 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 in. 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)> not equal 0, or vertical bar A(1/2) u(1)vertical bar(2) not equal A m(vertical bar A(1/2)u(0)vertical bar(2))vertical bar Au(0)vertical bar(2)), then the local solution is unique even if m is not Lipschitz continuous.

• 出版日期2010-3-1