Uniform stabilization for the semilinear wave equation in an inhomogeneous medium with locally distributed nonlinear damping and dynamic Cauchy–Ventcel type boundary conditions

Abstract

In this paper, we consider the Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a nonlinear damping distributed around a neighborhood [Formula: see text] of the boundary according to the Geometric Control Condition. Uniform decay rates of the associated energy are established and, in addition, the exact internal controllability for the linear problem is also proved. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [Contrôle Optimal des équations aux dérivées partielles. (2001); http://www.math.u-psud.fr/burq/articles/coursX.pdf ].

Uniform decay rate estimates for the 2D wave equation posed in an inhomogeneous medium with exponential growth source term, locally distributed nonlinear dissipation, and dynamic Cauchy–Ventcel–type boundary conditions

“…We also define the operator $$B:\mathcal {H}\rightarrow \mathcal {H}$$ given by $$$$\begin{equation} {B} = {\left( \def\eqcellsep{&}\begin{array}{cccc}0 & 0 & 0 & 0 \\[3pt] 0 & 0 & 0 & 0 \\[3pt] 0 & 0 & -a(x)g(\cdot ) & 0 \\[3pt] 0 & 0 & 0 & 0 \end{array} \right)} \end{equation}$$$$which is monotone, hemicontinuous, and bounded. The proof that these properties for operators $$A$$ and $$B$$ hold is the same as those obtained in [3].…”

“…The problem proposed in this paper is an extension for the works due to Almeida et al [3], Cavalcanti et al [11], and Simion Antunes et al [47] in a bidimensional domain. Our main goal is to prove the existence and uniqueness for solutions to problem (1.1) and, in addition, that these solutions decay uniformly to zero, that is, setting…”

“…The case 𝑝 ∈ [3,5) has been investigated by Dehman, Lebeau, and Zuazua in [24] and Joly and Laurent in [32]. The critical case 𝑝 = 5 has been studied in [34].…”

“…The problem proposed in this paper is an extension for the works due to Almeida et al. [3], Cavalcanti et al. [11], and Simion Antunes et al.…”

“…Alabau–Boussouira in [2] proved sharp or quasi‐optimal upper energy decay rates for a class of nonlinear feedbacks ranging from very weak nonlinear dissipation to polynomial or polynomial‐logarithmic decaying feedbacks at the origin. For the case of damped semilinear wave equations with a nonlinearity $$f$$, as considered in [3], with $$p\le 3$$, we reference [3] and the references presented therein (in particular, [22, 28, 45], and [51]). The case $$p\in \left[3,5\right)$$ has been investigated by Dehman, Lebeau, and Zuazua in [24] and Joly and Laurent in [32].…”