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The energy decay of divergence-free displacements for elastic waves with Neumann boundary condition

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  • 35-Xx Partial Differential Equations
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  • Communication
  • Ecology

Abstract

The energy decay of divergence-free displacements for elastic waves with Neumann boundary condition ∗ Rentaro Agemi & Hiroyuki Takamura Department of Complex Systems Future University-Hakodate 116-2 Kamedanakano-cho Hakodate, Hokkaido 041-8655, Japan e-mail : [email protected], [email protected] 1 Introduction It is well-known that the linear equation of displacements u = u(t, x) for isotropic, elastic waves is utt − div ( λ(div u)I + µ(∇u+ t∇u)) = 0, (1.1) where ∇u = (∂jui) is the gradient matrix and λ, µ are Lame´ constants which satisfy 3λ+2µ > 0, µ > 0 by physical requirements. The boundary condition for the traction problem is n · (λ(div u)I + µ(∇u+ t∇u)) = 0, (1.2) where n is an outer unit normal to the boundary. In this paper, we investigate divergence-free displacements 1 of the form; u(t, x) = t (x2ϕ(t, r),−x1ϕ(t, r), 0) , r = |x|, x = (x1, x2, x3). In the exterior domain {r ≥ b} with a constant b > 0, we find from (1.1) and (1.2) that ϕ satisfies the wave equation with a propergation speed c2 = ∗Partially supported by Northern Advancement Center for Science & Technology (No. H19-kyo-045). 1This is initiated by personal communications with Prof.S.Jimbo (Hokkaido Univ.). 1 √ µ, that is the equation of the transeverse wave, and Neumann boundary condition;  ϕtt − 4c 2 2 r ϕr − c22ϕrr = 0 in [0,∞)× [b,∞), ϕr = 0 on [0,∞)× {b}. (1.3) Here and hereafter we set the initial condition; ϕ(0, r) = ϕ0(r), ϕt(0, r) = ϕ1(r), (1.4) where ϕ0 and ϕ1 are given functions of bounded support. We also define the energy for (1.3) involving boundary integral; E(ϕ,R, t) = ∫ b≤|x|≤R { (rϕt) 2 + c22(2ϕ+ rϕr) 2 + 2c22ϕ 2 } dx + ∫ |x|=b 2bµϕ2dSx. (1.5) Then we can prove Theorem 1 Let ϕ be a solution of (1.3). 1. (energy conservation) The following equality is valid for all t > 0. E(ϕ,∞, t) = E(ϕ,∞, 0). (1.6) 2. (local energy decay) For any R > b, there exists a positive constant C = C(R) such that the following inequality is valid for all t > 0. tE

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