Global Existence Theorem for Nonlinear Wave Equation in Exterior Domain

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 ℝ 3 {{\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.

A blow-up result for a nonlinear damped wave equation in exterior domain: The critical case

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

Non-existence of weak solutions to nonlinear damped wave equations in exterior domains

Total energy decay for the wave equation in exterior domains with a dissipation near infinity

We study the decay estimates of the energy for the wave equation in an exterior domain with a localized dissipation. The dissipative term consists of the following two parts: The first part may be nonlinear and localized in a suitable bounded area, while the second part is linear in the outside of a big ball. So we may call such a dissipation ``half-linear" dissipation. We note that no geometrical condition is imposed on the boundary. As an application of the decay estimates we prove some global existence theorems for the wave equation with a nonlinear source term.

Local energy decay for linear wave equations with variable coefficients

Energy decay for linear dissipative wave equations in exterior domains

Nonlinear wave equations in exterior domains

Global existence theorem for nonlinear wave equation in exterior domain

Decay estimates for dissipative wave equations in exterior domains

A unique continuation principle and weak asymptotic behaviour of solutions to semilinear wave equations in exterior domains

Nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions

Combining the results in Ikehata-Matsuyama [5] with the Nakao inequality ([6], Lemma 2.2), we will derive more precise decay rate like E(t)≤C/(1+t)2 for the total energy E(t) to the mixed problem of the dissipative linear wave equation in an exterior domain through the multiplier method only.

We investigate the energy nondecay and existence of scattering states for solutions to the initial-boundary-value problem for the nonlinear wave equation in exterior domains. When the space dimension is odd, the domain meets no geometrical condition. Otherwise, we assume that the obstacle is convex. For odd-dimensional general domains, taking into account the effective dissipation in trapping regions, we can derive the existence of scattering states. In particular, we can obtain also an $L^2$ bound of solutions. The method in deriving the energy nondecay is to utilize Huyghens' principle. For even-dimensional domains outside the convex obstacle, the asymptotics stated in the odd-dimensional case are also valid.