Estimates of the rate of decay of solutions to the impedance mixed problem for the wave equation in a region with noncompact boundary

Abstract

The paper is devoted to the study of the behavior of the following mixed problem for large values of time: $$\begin{gathered} u_{tt} - \Delta u = 0, t > 0, x \in \Omega , u\left| {_{t = 0} = f} \right., u_t \left| {_{t = 0} = h} \right. \hfill \\ u\left| {_{x \in \Gamma _1 } } \right. = 0, \left( {\frac{{\partial u}}{{\partial \upsilon }} + g\left( x \right)\frac{{\partial u}}{{\partial t}}} \right)\left| {_{x \in } \Gamma _2 = 0,} \right. \hfill \\ \end{gathered} $$ where Ω is an unbounded region of ℝn with, generally speaking, noncompact boundary\(\partial \Omega = \Gamma _1 \cup \Gamma _2 \); the surface Γ is star-shaped (relative to the origin), ν is the unit outer normal to ∂Ω; and the initial functionsf andg are assumed to be sufficiently smooth and finite. Under certain restrictions on the part of the boundary Γ2 constrained by the impedance condition, we establish that one can match the impedanceg≥0 (characterizing the absorption of energy by the surface Γ2) to the geometric properties of this surface so that the energy on an arbitrary compact set will decay at a rate characteristic for the first mixed problem.