Abstract
We provide a polynomial decay rate for the energy of the wave equation with a dissipative boundary condition in a cylindrical trapped domain. A new kind of interpolation estimate for the wave equation with mixed Dirichlet-Neumann boundary condition is established from a construction based on a Fourier integral operator involving a good choice of weight functions.
Highlights
Introduction and main resultsIn this paper we study the behavior of the energy for the wave equation in a bounded cylindrical domain Ω in R3 with a boundary damping
We define by E (w, t) the energy of the solution w = w (x, t), E (w, t) ≡ |∂tw (x, t)|2 + |∇w (x, t)|2 dx
We argue locally around any point xo ∈ ωo
Summary
Where ν is the outward normal vector to ∂Ω and ∂ν is the normal derivative. We define by E (w, t) the energy of the solution w = w (x, t),. The first step in the proof of Theorem 1.2 consists in checking that we have the following observability estimate (see Theorem 2.1 in section 2): there exist C > 0 and T > 0 such that any solution u of (1.4) satisfies. We check after a lengthy but straightforward calculation that for any (x, t, s) ∈ R4 × (0, +∞), i∂s + h ∆ − ∂t2 (BP,Q (xo, ξo3) f ) (x, t, s) = 0 , and that for any (x1, x2, t, s) ∈ D × R × (0, +∞), On another hand, by multiplying (3.3) by u (x, t), solution of (1.5), and integrating by parts over Ω×R×[0, L], using (3.5), we have that for all (xo, ξo3) ∈ ωo ×(2Z + 1), for all (P, Q) ∈ N2 and all h ∈ Our goal consists in estimating separately the four integrals in (3.7), i.e., the four terms I1, I2, I3 and I4 in order to get the following result.
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