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Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface . We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain Ωcoupled with a thermoelastic plate equation defined on Γ0—a flat surface of the boundary \partial Ω . Thus, the coupling between the wave and the plate takes place on the interface Γ0. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the ``minimal amount'' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Ourmain result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ0, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural damping most often considered in the literature; (iii) there is no mechanical damping placed on the interface Γ0; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary \partial Ωare subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propagated onto the hyperbolic component of the structure. This is achieved by using a recently developed sharp theory of boundary traces corresponding to wave and plate equations, along with the analytic estimates recently established for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analyticity of the thermoelastic plate component allow one to establish appropriate inverse/ recovery type estimates which are critical for uniform stabilization. Our main result provides ``optimal'' uniform decay rates for the energy function corresponding to the full structure. These rates are described by a suitable nonlinear ordinary differential equation, whose coefficients depend on the growth of the nonlinear dissipation at the origin.\par http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics & Optimization Springer Journals

Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

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References (25)

Publisher
Springer Journals
Copyright
Copyright © Springer-Verlag New York Inc. 2000
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s002450010010
Publisher site
See Article on Publisher Site

Abstract

. We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain Ωcoupled with a thermoelastic plate equation defined on Γ0—a flat surface of the boundary \partial Ω . Thus, the coupling between the wave and the plate takes place on the interface Γ0. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the ``minimal amount'' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Ourmain result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ0, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural damping most often considered in the literature; (iii) there is no mechanical damping placed on the interface Γ0; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary \partial Ωare subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propagated onto the hyperbolic component of the structure. This is achieved by using a recently developed sharp theory of boundary traces corresponding to wave and plate equations, along with the analytic estimates recently established for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analyticity of the thermoelastic plate component allow one to establish appropriate inverse/ recovery type estimates which are critical for uniform stabilization. Our main result provides ``optimal'' uniform decay rates for the energy function corresponding to the full structure. These rates are described by a suitable nonlinear ordinary differential equation, whose coefficients depend on the growth of the nonlinear dissipation at the origin.\par

Journal

Applied Mathematics & OptimizationSpringer Journals

Published: Jan 1, 2000

Keywords: Structural acoustic models; Nonlinear boundary dissipation; Uniform stability; Thermoelastic plates; Free boundary conditions; AMS Classification. 35M10, 35B35.

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