Global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation (Q616650)

The Kirchhoff equation describes the oscillations of initially stretched elastic strings. The displacement of the string thus depends only one space variable and a time variable. It satisfies a nonlinear second-order hyperbolic integro-differential equation. The author considers an \(n\)-dimensional generalization of this equation defined over an exterior open bounded domain in \(n\)-dimensional Euclidean space. It is assumed that the displacement vanishes at the boundary of the domain. The author proves by resorting to rather ingenious functional analysis techniques that the Kirchhoff equation admits a unique global solution provided that the initial values belong to certain Sobolev spaces of fractional order and satisfy some norm inequalities. The properties of the linear wave equation, generalized Fourier transform and the resolvent operator are amply used in the analysis.