Exponential Stability of a Thermoelastic System with Free Boundary Conditions without Mechanical Dissipation

Abstract

Abstract. We show herein the uniform stability of a thermoelastic plate model with no added dissipative mechanism on the boundary (uniform stability of a thermoelastic plate with added boundary dissipation was shown in [J. LAGNESE, Boundary Stabilization of Twin Plates, SIAM Stud. Appl. Math. 10, SIAM, Philadelphia, PA, 1989], as was that of the analytic case-where rotational forces are neglected-in [Z. LIU and S. ZHENG, Quarterly Appl. Math., 55 (1997), pp. 551-564]). The proof is constructive in the sense that we make use of a multiplier with respect to the coupled system involved so as to generate a fortiori the desired estimates; this multiplier is of an operator theoretic nature, as opposed to the more standard differential quantities used for related work. Moreover, the particular choice of our multiplier becomes clear only after recasting the PDE model into an associated abstract evolution equation.

Existence and Exponential Decay of Solutions to a Quasilinear Thermoelastic Plate System

“…And in fact this is indeed the case in linear models, where exponential decay rates for the linear energy have been established [4,5,27] for linear thermoelastic plates and more recently in [8,13] for semilinear plates. The situation in the quasilinear case is much more complex, due to the unboundedness of the nonlinear term with respect to the topology induced by the energy.…”

“…We apply the method of multipliers introduced by Avalos and Lasiecka [5] for linear thermoelastic problems. The method uses two multipliers…”

Long-time dynamics of two classes of beam and plate equations

“…In fact, still speaking of a Von Karman scalar model, one obtains uniform convergence to an attracting set without any mechanical dissipation for the model that accounts additionally rotational inertia (the term γ∆w tt is added to the plate equation) [6]. This latter effect provides regularizing effect on the velocity of oscillations.…”

“…Full theory of a long-time behavior (including theory of attractors) has been developed for a thermoelastic von Karman scalar equation in the variables (w, θ ), that is the model (1.9) with thermal effects, with either clamped or hinged boundary conditions [20] and without any mechanical dissipation. The analysis in [20] relies on techniques developed in [6,7] that includes special nonlocal multipliers along with quisi-stability theory developed in [19]. In the case of clamped or hinged boundary conditions one also shows that the estimates for the dimension of the attracting set are independent on rotational inertia (γ∆w tt ).…”

“…This stabilization is uniform with respect to rotational inertia parameter γ 0. The method of obtaining estimates relies on multipliers developed in [6] and also [41]. However, this methods fail when free boundary conditions are imposed.…”