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

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 Ω coupled 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? Our main 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 Ω are 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

[1]  Irena Lasiecka,et al.  Uniform decay rates for full von karman system of dynamic theromelasticity with free boundary conditions and partial boundary dissipation , 1999 .

[2]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[3]  George Avalos,et al.  THE EXPONENTIAL STABILITY OF A COUPLED HYPERBOLIC/PARABOLIC SYSTEM ARISING IN STRUCTURAL ACOUSTICS , 1996 .

[4]  Irena Lasiecka,et al.  The Strong Stability of a Semigroup Arising from a Coupled Hyperbolic/Parabolic System , 1998 .

[5]  Irena Lasiecka,et al.  Differential riccati equation for the active control of a problem in structural acoustics , 1996 .

[6]  A. Balakrishnan.  Applied Functional Analysis , 1976 .

[7]  J. Lagnese,et al.  Boundary Stabilization of Linear Elastodynamic Systems , 1983 .

[8]  Irena Lasiecka,et al.  Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation , 1998 .

[9]  Bo Liu,et al.  On the Spectral Properties and Stabilization of Acoustic Flow , 1998, SIAM Journal on Applied Mathematics.

[10]  S. Mitter,et al.  Representation and Control of Infinite Dimensional Systems , 1992 .

[11]  Mitsuru Ikawa,et al.  Mixed problems for hyperbolic equations of second order , 1968 .

[12]  D. Russell.  Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions , 1978 .

[13]  Günter Leugering,et al.  Uniform stabilization of a nonlinear beam by nonlinear boundary feedback , 1991 .

[14]  I. Lasiecka.  Existence and uniqueness of the solutions to second order abstract equations with nonlinear and nonmonotone boundary conditions , 1994 .

[15]  J. Lagnese.  Boundary Stabilization of Thin Plates , 1987 .

[16]  Daniel Tataru,et al.  ON THE REGULARITY OF BOUNDARY TRACES FOR THE WAVE EQUATION , 1998 .

[17]  Irena Lasiecka,et al.  Sharp Regularity Theory for Elastic and Thermoelastic Kirchoff Equations with Free Boundary Conditions , 2000 .

[18]  L. Markus,et al.  Stabilization of a hybrid system of elasticity by feedback boundary damping , 1988 .

[19]  F. Fahroo,et al.  A new model for acoustic-structure interaction and its exponential stability , 1999 .

[20]  J. Lagnese.  Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary , 1990 .

[21]  Irena Lasiecka,et al.  Boundary stabilization of a 3-dimensional structural acoustic model , 1999 .

[22]  Irena Lasiecka,et al.  Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping , 1993, Differential and Integral Equations.

[23]  V. Barbu.  Nonlinear Semigroups and di erential equations in Banach spaces , 1976 .