Polynomial Decay Rate for Dissipative Wave Equations with Mixed Boundary Conditions
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Polynomial Decay Rate for Dissipative Wave Equations with Mixed Boundary Conditions K. Laoubi1 · D. Seba1
Received: 18 July 2019 / Accepted: 25 January 2020 © Springer Nature B.V. 2020
Abstract By using Fourier-Bessel analysis combined with Ingham’s inequality and the multiplier techniques, a polynomial decay rate for the energy of the wave equation, in a two-dimensional bounded domain with Wentzell-Dirichlet boundary conditions, is established. Keywords Stabilization · Wave equation · Energy decay rates · Wentzell’s boundary conditions · Fourier-Bessel analysis · The multiplier method · Semigroup theory Mathematics Subject Classification (2010) 35L05 · 35B35 · 35B40
1 Introduction In the 1950s Feller [20, 21] and Wentzell [37] initiated The study of evolution equations with Wentzell’s boundary conditions which were subsequently developed and used in many papers; see, for example [7, 11, 16, 26, 28, 31]. Wentzell’s boundary conditions arise in the context of multidimensional diffusion processes. They can also be derived as artificial boundary conditions in exterior problems or approximate boundary conditions in asymptotic problems [9, 15] and in modeling climatology [18]. For the role of Wentzell boundary conditions in linear and nonlinear analysis, we refer the reader to [17]. In this work, we are concerned by the stabilization of the wave equation on a ring subject to Wentzell-Dirichlet boundary conditions. Because the resolvent operator of this equation is unbounded on the imaginary axis [23], we have to find a weaker decay of the energy. Clearly we provide methods and sufficient conditions able to guarantee a polynomial decay of the
B D. Seba
[email protected] K. Laoubi [email protected]
1
Dynamic of Engines and Vibroacoustic Laboratory, Faculty of Engineer’s Sciences, Boumerdes University, Boumerdes, Algeria
K. Laoubi, D. Seba
energy of the considered problem. The proof relies on a Fourier-Bessel analysis, spectral theory, the multiplier technique and a method combined with Ingham’s inequality [35]. There exist several time domain methods for the polynomial decay rate estimation in the literature, you can find a semigroup theory approach in [4, 6, 10, 12], an energy inequality approach established in [2, 13, 32, 36]. By means of a proper Carleman estimate, a logarithmic decay rate of the energy was established in [15], a Fourier analysis method in [29, 40], a perturbation method in [22, 41], an optimal observability inequality in [19], a method based on the order of unboundedness of the resolvent operator on the imaginary axis in [30], an another method using Ingham’s inequality can be found in [5] and the references contained therein. The second section, of this paper, presents some basic concepts assumption and preliminary results to prove the main results. The third section is where we analyze the onedimensional wave dissipative equation depending on a parameter. The last section establishes the main result consisting of the polynomial stability for the solution of our problem.
2 The Problem
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