Uniform stabilization of the higher-dimensional system of thermoelasticity with a nonlinear boundary feedback

Abstract

Abstract.Using multiplier techniques and Lyapunov methods, we derive explicit decay rates for the energy in the higher-dimensional system of thermoelasticity with a nonlinear velocity feedback on part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary.

Stabilization of a system of anisotropic thermoelasticity by nonlinear boundary and internal feedbacks

“…we obtain decays similar to the ones from Theorem 2.3 of [16]. Indeed, if p = α = 1, then Ψ −1 (t) = e −t and we conclude an exponential decay.…”

“…The proof of this lemma is similar to the one of Lemma 3.2 of [16]. The theory of nonlinear semi-groups [21] leads to Theorem 2.1.…”

“…Recently, Liu and Zuazua [16] have established, still in the case f = 0, exponential, polynomial or logarithmic decay of the energy for some nonlinearities g. The aim of this work is to generalize these results to the case of an anisotropic system, to the case f = 0 and to the case of natural boundary conditions. This last case answers a question raised in [16]. For this purpose, in the linear case we establish integral inequalities as in [15] and [1] leading to the exponential decay and in the nonlinear case, we use the theorical results established in [13].…”

“…The stabilization of different variants of the system (3) has been studied in the literature, notably in [15,16] (see also [5,17,18,19,20]) in the isotropic case but with nonnatural boundary conditions, namely by replacing in (3) the term σ(u) · ν by µ ∂u ∂n + (λ + µ)νdiv u. In [15], Liu considered the case f = 0 and the linear feedback, i.e., g(x) = x.…”

“…In [15], Liu considered the case f = 0 and the linear feedback, i.e., g(x) = x. Recently, Liu and Zuazua [16] have established, still in the case f = 0, exponential, polynomial or logarithmic decay of the energy for some nonlinearities g. The aim of this work is to generalize these results to the case of an anisotropic system, to the case f = 0 and to the case of natural boundary conditions. This last case answers a question raised in [16].…”