Global Existence Theorem for Nonlinear Wave Equation in Exterior Domain

Abstract

The global existence of solutions for the nonlinear wave equation has been extensively studied. For the Cauchy problem, Klainerman has made a remarkable improvement recently. That is, he showed that if the spatial dimension is not smaller than 6 and initial data are small and smooth; the Cauchy problem for the fully nonlinear wave equation has a unique classical global solution. On the other hand it is important to consider the initial boundary value problem for the nonlinear wave equation in an exterior domain in order to study scattering of a reflecting object for the nonlinear wave equation. The chapter presents the proof that if the spatial dimension is not smaller than 3 and initial data are small and smooth, the global unique existence theorem of classical solutions for a large class of nonlinear wave equations in exterior domains with the homogeneous Dirichlet boundary condition is obtained. It includes the nonlinear vibration equation.