Stability results of locally coupled wave equations with local Kelvin-Voigt damping: Cases when the supports of damping and coupling coefficients are disjoint

Abstract

In this paper, we study the direct/indirect stability of locally coupled wave equations with local Kelvin-Voigt dampings/damping, where we assume that the supports of the dampings and the coupling coefficients are disjoint. First, we prove the well-posedness, strong stability, and polynomial stability for some one dimensional coupled systems. Moreover, under some geometric control conditions, we prove the well-posedness and strong stability in the multi-dimensional case.

Appendix A. Some notions and stability theorems

In order to make this paper more self-contained, we recall in this short appendix some notions and stability results used in this work.

Definition A.1

Assume that A is the generator of $C_0-$semigroup of contractions $\left( e^{tA}\right) _{t\ge 0}$ on a Hilbert space H. The $C_0-$semigroup $\left( e^{tA}\right) _{t\ge 0}$ is said to be

• (1) Strongly stable if

  $$\begin{aligned} \lim _{t\rightarrow +\infty } \Vert e^{tA}x_0\Vert _H=0,\quad \forall \, x_0\in H. \end{aligned}$$

• (2) Exponentially (or uniformly) stable if there exists two positive constants M and $\varepsilon$ such that

  $$\begin{aligned} \Vert e^{tA}x_0\Vert _{H}\le Me^{-\varepsilon t}\Vert x_0\Vert _{H},\quad \forall \, t>0,\ \forall \, x_0\in H. \end{aligned}$$

• (3) Polynomially stable if there exists two positive constants C and $\alpha$ such that

  $$\begin{aligned} \Vert e^{tA}x_0\Vert _{H}\le Ct^{-\alpha }\Vert x_0\Vert _{H},\quad \forall \, t>0,\ \forall \, x_0\in D(A). \end{aligned}$$

  $\square$

To show the strong stability of the $C_0$-semigroup $\left( e^{tA}\right) _{t\ge 0}$ we rely on the following result due to Arendt and Batty (1988):

Theorem A.2

Assume that A is the generator of a C$_0-$semigroup of contractions $\left( e^{tA}\right) _{t\ge 0}$ on a Hilbert space H. If A has no pure imaginary eigenvalues and $\sigma \left( A\right) \cap i{\mathbb {R}}$ is countable, where $\sigma \left( A\right)$ denotes the spectrum of A, then the $C_0$-semigroup $\left( e^{tA}\right) _{t\ge 0}$ is strongly stable. $\square$

Concerning the characterization of polynomial stability stability of a $C_0-$semigroup of contraction $\left( e^{tA}\right) _{t\ge 0}$ we rely on the following result due to Borichev and Tomilov (2010) (see also Batty and Duyckaerts 2008 and Liu and Rao 2005):

Theorem A.3

Assume that A is the generator of a strongly continuous semigroup of contractions $\left( e^{tA}\right) _{t\ge 0}$ on ${\mathcal {H}}$. If $i{\mathbb {R}}\subset \rho (A)$, then for a fixed $\ell >0$ the following conditions are equivalent:

$$\begin{aligned} \limsup _{{\lambda }\in \mathbb R, \ |{\lambda }| \rightarrow \infty }\frac{1}{|{\lambda }|^\ell }\left\| (i{\lambda }I-A)^{-1}\right\| _{{\mathcal {L}}({\mathcal {H}})}<\infty , \end{aligned}$$

(A.1)

$$\begin{aligned} \Vert e^{tA}U_{0}\Vert ^2_{\mathcal H} \le \frac{C}{t^{\frac{2}{\ell }}}\Vert U_0\Vert ^2_{D(A)},\forall t>0, U_0\in D(A), \text {for some} C>0. \end{aligned}$$

(A.2)

$\square$