Space-time estimate for the perturbed linear elastic wave equations exterior to a ball with radial data and its application

Abstract In this paper we are devoted to establishing a kind of weighted space-time estimate for the perturbed linear elastic wave equations in the exterior domain outside a ball in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> {{\mathbb{R}}^{3}} , assuming the initial data are radial symmetric. This solves an interesting open problem posed in [J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z. 256 2007, 3, 521–549] in the radial symmetric case. As an application, we prove the almost global existence for the nonlinear elastic wave equations in the domain exterior to a ball with inhomogeneous boundary condition and radial symmetric initial data.