Global well-posedness of Kirchhoff systems

Abstract

The aim of this paper is to establish the $H^1$ global well-posedness for Kirchhoff systems. The new approach to the construction of solutions is based on the asymptotic integrations for strictly hyperbolic systems with time-dependent coefficients. These integrations play an important role to setting the subsequent fixed point argument. The existence of solutions for less regular data is discussed, and several examples and applications are presented.Comment: 24 page

Necessary condition on the dissipative term for the asymptotically free property of dissipative Kirchhoff equations

“…In the case b ≡ 0 the unique global solvability of (1.1) and (1.2) (and (1.3) if ∂Ω ̸ = ∅) in Sobolev spaces is investigated by many authors, but is known only for small data in some class (for example, see [7,[2][3][4]15,18,13] in the whole space, and [14,8,21,22,10], in exterior domains in R n ). We prepare a class for the global solvability.…”

Scattering for strictly hyperbolic systems with time‐dependent coefficients

“…Dispersive estimates in these settings are based on the multi‐dimensional version of the van der Corput lemma established in 12, 13. Optimal dispersion and Strichartz estimates for hyperbolic systems with time‐dependent coefficients will be discussed in 9, 10 and will appear elsewhere, as well as the applications to Kirchhoff systems of the results obtained there and in the present paper.…”

On t-dependent hyperbolic systems. Part 2

“…endowed with suitable initial conditions 14) and for coefficient functions a k,α ∈ T {m − k − |α|}. Assuming that the principal part is uniformly strictly hyperbolic, i.e., that the polynomial…”