Almost global well-posedness of Kirchhoff equation with Gevrey data

Abstract The aim of this note is to present the almost global well-posedness result for the Cauchy problem for the Kirchhoff equation with large data in Gevrey spaces. We also briefly discuss the corresponding results in bounded and in exterior domains.


Introduction
In this paper, we consider the Cauchy problem for the Kirchhoff equation: ( This equation has a long history, having appeared as a model for the string motion in the book [10] of Kirchhoff.Bernstein [2] proved for it the existence of global in time analytic solutions on an interval of the real line.Arosio and Spagnolo [1] discussed analytic solutions in higher spatial dimensions, and in [4] D'Ancona and Spagnolo proved analytic well-posedness for the degenerate Kirchhoff equation, with a related work by Kajitani and Yamaguti [9].In the quasi-analytic class, it is known that global solutions exist, see [7,15,16].Moreover, Manfrin [11] showed the existence of time global solutions in Sobolev spaces corresponding to non-analytic data having a spectral gap, see also [8].For the existence of local solutions in Gevrey spaces, we refer the reader to the results of Ghisi and Gobbino (see [6,5]).
For small data, many more results are available.Here we refer the reader to [12], where a detailed literature review for solutions to (2) is given, as well as the most recent results on small data.In this note, we are interested in the existence of solutions for large data, so we only refer to [13] for a detailed survey dealing with large, but especially with small data.The results announced in this note will appear in [14].

Statement of results
In this section, we discuss the existence of solutions to (2) for Gevrey data.Let us recall the definition of the Gevrey class of L 2 type.For s ≥ 1, we denote by Here f (ξ ) is the Fourier transform of f (x).The class γ s L 2 is endowed with the inductive limit topology.In particular, if s = 1, .
A special feature of the equation ( 2) is its Hamiltonian structure: for the energy we have on the interval of the existence of solutions.
The following result gives the almost global existence for Gevrey solutions for (2) with large Gevrey data.
We note that the smallness of data is not required in Theorem 1: the size of the data is measured by constants M and R that are allowed to be large.However, an interesting feature is that the regularity of the data is related to the size, although this regularity is measured within the same Gevrey class γ s L 2 .So, we can informally describe conditions of Theorem 1 by saying that 'the larger the data is the more regular it has to be'.
The statement concerning the regularity in the class γ s L 2 can be refined.Namely, we have the following remark.
Remark 2. Assume the conditions of Theorem 1. Then the solution u satisfies u ∈ We conclude this note by giving an outline of the proof of Theorem 1.To begin with, we consider the linear Cauchy problem with the same data (u 0 , u 1 ) as in the nonlinear Cauchy problem (2).The coefficient c(t) is assumed to be of Lipschitz regularity on [0, T ], and satisfy the conditions for some M > 2 and K > 0.Then, after using the energy estimates for solutions to (3) (in fact, a refined version of energy estimates from Colombini, Del Santo and Kinoshita [3]), and choosing the constant K such that where we put Thanks to the continuity argument, we can also prove that Thus, the Schauder-Tychonoff fixed point theorem allows us to conclude that the solution v of the linear Cauchy problem (3) with data (u 0 , u 1 ) is also the solution to the nonlinear Cauchy problem (2).
The argument in the proof of Theorem 1 is also applicable to the initial-boundary value problems in an open set of R n : bounded domains and exterior domains with analytic boundary.The results can be proved by the Fourier series expansion method in bounded domains, and by the generalised Fourier transform method in exterior domains, respectively.It is known from the spectral theorem that a self-adjoint operator on a separable Hilbert space is unitarily equivalent to a multiplication operator on some L 2 (M, μ), where (M, μ) is a measure space.Then L 2 ( ) is unitarily equivalent to L 2 (R n ).
This means that the Fourier transform method in R n is available for the L 2 space on an open set in R n ; any multiplier acting on L 2 (R n ) is unitarily transformed into a multiplier acting on L 2 ( ).For more details, we refer the reader to [14].