Stabilization of two strongly coupled hyperbolic equations in exterior domains

TL;DR

This paper investigates the energy decay and boundedness of solutions for two coupled hyperbolic equations in exterior domains, demonstrating uniform decay and exponential energy decay under certain damping and coupling conditions.

Contribution

It establishes conditions under which the total energy decays uniformly and the solutions remain bounded, including exponential decay for coupled Klein-Gordon equations with equal speeds.

Findings

Total energy decays uniformly when damping includes the coupling set.

The $L^2$-norm of solutions remains bounded under specified conditions.

Exponential energy decay occurs for coupled Klein-Gordon equations with equal speeds.

Abstract

In this paper we study the behavior of the total energy and the $L^2$-norm of solutions of two coupled hyperbolic equations by velocities in exterior domains. Only one of the two equations is directly damped by a localized damping term. We show that, when the damping set contains the coupling one and the coupling term is effective at infinity and on captive region, then the total energy decays uniformly and the $L^2$-norm of smooth solutions is bounded. In the case of two Klein-Gordon equations with equal speeds we deduce an exponential decay of the energy.

Equations

$$\{\begin{array}{rlr}& {\partial }_t^{2}u-\Delta u+mu+a(x){\partial }_{t}u=0& \text{ in }{\mathbb{R}}_{+}\times \Omega ,\\ & u=0& \text{ on }{\mathbb{R}}_{+}\times \Gamma ,\\ & (u(0,.),{\partial }_{t}u(0,.))=(u_0,u_1)& in\Omega ,\end{array}$$

$$E_u(t)=\frac{1}{2}{\int }_{\Omega }|{\partial }_{t}u(t,x){|}^2+|\nabla u(t,x){|}^2+m|u(t,x){|}^{2}dx,$$

$$E_u(0)={\int }_0^t{\int }_{\Omega }a(x)|{\partial }_{t}u(t,x){|}^{2}dxdt+E_u(t),t>0.$$

$$E_u(t)⩽C(1+t{)}^{-1-d(\frac{1}{i}-\frac{1}{2})}{I}_i^2,$$

$$\Vert u(t,.){\Vert }_{L^2}^2⩽C(1+t{)}^{d(\frac{1}{i}-\frac{1}{2})}{I}_i^2,$$

$$E_u(t)⩽C(1+t{)}^{-1}{I}_2^{2}and\Vert u(t){\Vert }_{L^2}^2⩽C{I}_2^2,forallt⩾0.$$

$$E_u(t)⩽C{e}^{-\delta t}{E}_u(0),forallt⩾0,$$

$$\{\begin{array}{rlr}& {\partial }_t^{2}u-\Delta u+m_{1}u+b(x){\partial }_{t}v+a(x){\partial }_{t}u=0& \text{ in }{\mathbb{R}}_{+}\times \Omega ,\\ & {\partial }_t^{2}v-{\gamma }^2\Delta v+m_{2}v-b(x){\partial }_{t}u=0& \text{ in }{\mathbb{R}}_{+}\times \Omega ,\\ & u=v=0& \text{ on }{\mathbb{R}}_{+}\times \Gamma ,\\ & (u(0,.),{\partial }_{t}u(0,.))=(u_0,u_1)& in\Omega ,\\ & (v(0,.),{\partial }_{t}v(0,.))=(v_0,v_1)& in\Omega ,\end{array}$$

$$E_{u,v}(t)$$

$$+\frac{1}{2}{\int }_{\Omega }{\gamma }^2|\nabla v(t,x){|}^2+|{\partial }_{t}v(t,x){|}^2+m_2|v(t,x){|}^{2}dx.$$

$$\displaystyle\|(w_{0},w_{1},w_{2},w_{3})\|_{\mathcal{H}}=\Big{(}\int_{\Omega}|\nabla w_{0}|^{2}+\gamma^{2}|\nabla w_{2}|^{2}+m_{1}|w_{0}|^{2}+m_{2}|w_{2}|^{2}+|w_{1}|^{2}+|w_{3}|^{2}\ dx\Big{)}^{\frac{1}{2}}.$$

$$\{\begin{array}{rl}& {\mathcal{U}}_t+\mathcal{A}\mathcal{U}=0,\\ & \mathcal{U}(0)={\mathcal{U}}_0\in \mathcal{H},\end{array}$$

$$\mathcal{U}=\left(\begin{array}{c}u\\ {\partial }_{t}u\\ v\\ {\partial }_{t}v\text{par}\end{array}\right),{\mathcal{U}}_0=\left(\begin{array}{c}{u}_0\\ u_1\\ v_0\\ v_1\text{par}\end{array}\right)$$

$$D(\mathcal{A})=\{\mathcal{U}\in \mathcal{H},\mathcal{A}\mathcal{U}\in \mathcal{H}\}$$

$$\mathcal{A}=\left(\begin{array}{cccc}0& -Id& 0& 0\\ -\Delta +m_{1}Id& a& 0& b\\ 0& 0& 0& -Id\\ 0& -b& -{\gamma }^2\Delta +m_{2}Id& 0\end{array}\right).$$

$$\frac{d}{dt}{E}_{u,v}(t)=-{\int }_{\Omega }a(x)|{\partial }_{t}u(t,x){|}^{2}dx.$$

$$E_{u,v}(t)⩽M{e}^{-\beta t}{E}_{u,v}(0),forallt⩾0,$$

$$\{\begin{array}{rlr}& {\partial }_t^{2}u(t,x)-\Delta u(t,x)+b(x)v(t,x)+a(x){\partial }_{t}u(t,x)=0& \text{ in }{\mathbb{R}}_{+}\times \Omega ,\\ & {\partial }_t^{2}v(t,x)-\Delta v(t,x)+b(x)u(t,x)=0& \text{ in }{\mathbb{R}}_{+}\times \Omega ,\\ & u=v=0& \text{ on }{\mathbb{R}}_{+}\times \Gamma ,\\ & (u(0,.),{\partial }_{t}u(0,.))=(u_0,u_1)& in\Omega ,\\ & (v(0,.),{\partial }_{t}v(0,.))=(v_0,v_1)& in\Omega ,\end{array}$$

$$O\subset B_0=\{x\in {\mathbb{R}}^d,|x|<R_0\}.$$

$$T_{\omega }=\inf \{T>0,(\omega ,T)\text{ satisfies GCC}\}.$$

$$I_{\gamma }^2=E_{u,v}(0)+(1-\frac{1}{{\gamma }^2}{)}^2{E}_{{\partial }_{t}u,{\partial }_{t}v}(0)+\Vert (u,v)(0){\Vert }_{L^2(\Omega )}^2$$

$${\mathcal{H}}_{\gamma }=\{\begin{array}{l}\mathcal{H}\cap (L^2(\Omega ){)}^4,if\gamma =1,\\ D(\mathcal{A})\cap (L^2(\Omega ){)}^4,if\gamma \ne 1.\end{array}$$

$$E_{u,v}(t)⩽C(1+t{)}^{-1}{I}_{\gamma }^2\text{ and}\Vert (u,v)(t){\Vert }_{L^2}^2⩽C{I}_{\gamma }^2,\text{ for all }t⩾0,$$

$$E_{u,v}(t)⩽C{e}^{-\alpha t}{E}_{u,v}(0),\text{ for all }t⩾0,$$

$$E_{u,v}(t)⩽\frac{C}{t^n}\sum _{k=0}^n{E}_{{\partial }_t^{k}u,{\partial }_t^{k}v}(0),\text{ for all }t⩾0,$$

$$E^R(u,v,t)$$

$$+\frac{1}{2}{\int }_{|x|>R}|{\partial }_{t}v(t,x){|}^2+{\gamma }^2|\nabla v(t,x){|}^2+m_2|v(t,x)|dx,$$

$$E_R(u,v,t)$$

$$+\frac{1}{2}{\int }_{{\Omega }_R}|{\partial }_{t}v(t,x){|}^2+{\gamma }^2|\nabla v(t,x){|}^2+m_2|v(t,x)|dx,$$

$$\Vert (u,v)(t){\Vert }_{L^2(|x|>R_2)}^2+{\int }_0^t{E}^{R_2}(u,v,s)ds\lesssim C_{\epsilon }(E_{u,v}(0)+(1-\frac{1}{{\gamma }^2}{)}^2{E}_{{\partial }_{t}u,{\partial }_{t}v}(0))$$

$$+\epsilon {\int }_0^t{E}_{u,v}(s)ds+C_{\epsilon }{\int }_0^t{\int }_{{\Omega }_{R_2}}|u{|}^2+|v{|}^{2}dxds+\Vert (u,v)(0){\Vert }_{L^2(\Omega )}^2,$$

$$\varphi(x)=\begin{cases}1&\text{ for }|x|\geqslant R_{2}\\ 0&\text{ for }|x|\leqslant R_{1}.\end{cases}\$$

Full text

Email : lassaad.aloui@fst.utm.tn.

houdaazaza@gmail.com.

1. Introduction and statement of the results

Let $\Omega$ be a domain of $\mathbb{R}^{d}$ ,$d\geqslant 2$. We denote by $\Delta$ the Laplace operator on $\Omega$ with Dirichlet boundary condition. We consider the following hyperbolic equation with localized linear damping

[TABLE]

where $a\in L^{\infty}(\Omega)$ is a nonnegative smooth function and $m\in\mathbb{R}_{+}$. It is easy to verify that the energy given by

[TABLE]

is non-increasing and

[TABLE]

When $m=0$, the stabilization problem for the linear damped wave equation has been studied by several authors. More precisely, when $\Omega$ is bounded, the uniform decay of the total energy is equivalent to the geometric control condition of Bardos et al [7]. On the other hand, if $\Omega$ is not bounded then, in general, the decay rate of the total energy cannot be uniform. Indeed, in the whole space,i.e. $\Omega=\mathbb{R}^{d}$, Matsumura [19] obtained a precise $L^{p}-L^{q}$ type decay estimate for solutions of $\eqref{pbonde}$, when $a(x)=1$,

[TABLE]

[TABLE]

where $C$ is a positive constant, $i\in[1,2]$ and $I_{i}^{2}=\|u_{0}\|_{H^{1}}^{2}+\|u_{1}\|_{L^{2}}^{2}+\|u_{0}\|_{L^{i}}^{2}+\|u_{1}\|_{L^{i}}^{2}$. The proof in [19] is based on a Fourier transform method. In the case of exterior domains and when $a(x)\geqslant a^{-}>0$ on $\Omega$, it is easy to show that the weak solution $u$ of the system $\eqref{pbonde}$ satisfies

[TABLE]

In [20], Nakao obtained the estimate $\eqref{0101}$ for a damper which is positive near infinity and near a part of the boundary (Lions’s condition). Daoulatli in [11] generalized this result by assuming that each trapped ray meets the damping region which is also effective at infinity. Recently, Aloui et al [6] established the uniform stabilization of the total energy for the system $\eqref{pbonde}$ when the initial data are compactly supported. They proved that the rate of decay turns out to be the same as those of the heat equation, which shows that the effective damper at space infinity strengthens the parabolic structure in the equation.

In the case $m>0$, the energy $\eqref{eneg11}$ contains the $L^{2}$ norm. Then, using the semi-group property, the type of decay $\eqref{0101}$ implies the expnential one

[TABLE]

where $C,\delta$ positive constants. In [23] Zuazua considered the nonlinear Klein-gordon equations with dissipative term and he proved the exponential decay of energy through the weighted energy method. This result has been generalized by Aloui et al [5] for more general nonlinearities. We refer the reader to the works of Dehman et al [9] and Laurent et al [14] for related results.

In this paper we will study the stabilization problem for a system of two coupled hyperbolic equations on exterior domain. More precisely, let $O$ be a compact domain of $\mathbb{R}^{d}$ with $\mathcal{C}^{\infty}$ boundary $\Gamma=\partial O$ and $\Omega=\mathbb{R}^{d}\backslash O$

[TABLE]

where $b\in L^{\infty}(\Omega)$ is a smooth function, $m_{1},m_{2}\in\mathbb{R}_{+}$ and $\gamma$ is a positive constant.

We associate to the system $\eqref{pb311}$ the energy functional given by

[TABLE]

Let \mathcal{H}=\Big{(}H_{D}^{1}(\Omega)\times L^{2}(\Omega)\Big{)}^{2} be the completion of $(C_{0}^{\infty}(\Omega))^{4}$ with respect to the norm

[TABLE]

The linear evolution equation $\eqref{pb311}$ can be rewritten under the form

[TABLE]

where

[TABLE]

and the unbounded operator $\mathcal{A}$ on $\mathcal{H}$ with domain

[TABLE]

is defined by

[TABLE]

From the linear semi-group theory, we can infer that for $\mathcal{U}_{0}\in\mathcal{H}$ the problem $\eqref{pb32}$ admits a unique solution $\mathcal{U}\in C^{0}([0,+\infty[,\mathcal{H})$.

In addition, if $\mathcal{U}_{0}\in D(\mathcal{A}^{n})$, for $n\in\mathbb{N}$, then the solution $\mathcal{U}\in\displaystyle\bigcap_{i=0}^{n}C^{n-i}(\mathbb{R}_{+},D(\mathcal{A}^{i}))$.

It is easy to verify that

[TABLE]

Thus $E_{u,v}(t)$ is decreasing with respect to time.

In bounded domain and under some geometric conditions, Kapitonov [13] considered the case of equal speeds ($\gamma=1$) and proved the uniform decay

[TABLE]

where $M,\beta>0$. In [3], Ammar et al studied the indirect stability of system (1.7) in the case of one-dimensional space and when $a$ and $b$ have disjoint supports. More precisely, they established that the ”classical” internal damping applied to only one of the equations never gives exponential stability if $\gamma\neq 1$ and for the case $\gamma=1$ they gave an explicit necessary and sufficient conditions for the stability to occur. In [22], Toufayli generalized this result for different speeds and established, under some geometric conditions, a polynomial stability.

The problem of the indirect stabilization has been also studied for coupled wave equations by displacements (weakly coupled). Indeed Alabau et al [1] considered the following system

[TABLE]

where $\Omega$ is a bounded domain. They proved that the system $\eqref{pb3333}$ can not be exponentially stable and when the coupling term is constant they established a polynomial decay. In [2] Alabau et al improved this result by assuming that the regions $\{a>0\}$ and $\{b>0\}$ both verify GCC and the coupling term satisfies a smallness assumption. This result has been generalized by Aloui et al [4], for more natural smallness condition on the infinity norm of the coupling term. Recently, Daoulatli [10] showed that the rate of energy decay for solutions to the system on a compact manifold with a boundary is determined from a first order differential equation when the coupling zone and the damping zone verify the GCC.

In the sequel, we fix a constant $R_{0}>0$ such that

[TABLE]

Suppose that there exist two positive constants $a^{-}$ and $b^{-}$ such that the damping set $\omega_{a}:=\{a(x)>a^{-}>0\}$ and the coupling set $\omega_{b}:=\{b(x)>b^{-}>0\}$ are non-empty open subsets of $\Omega$. As usual for damped wave (resp. Klein-Gordon) equations, we have to make some geometric assumptions on the sets $\omega_{a}$ and $\omega_{b}$ so that the energy of a single wave decays sufficiently rapidly at infinity. Here, we shall use the Geometric control condition.

Definition 1.1**.**

(see [7, 15]) We say that a set $\omega$ of $\Omega$ satisfies the geometric control condition GCC if there exists $T>0$ such that from every point in $\Omega$ the generalized geodesic meets the set $\omega$ in a time $t<T$.

If $\omega$ satisfies GCC, we set

[TABLE]

We need also the following assumptions

• $(\mathcal{A}_{1})$

$supp(b)\ \subset\ supp(a)$.

• $(\mathcal{A}_{2})$

There exists $R_{1}>R_{0}$ such that

• $\bullet$

$B_{R_{1}}^{c}\subset\omega_{a}\cap\omega_{b}$, if $\ (m_{1},m_{2})\in\mathbb{R}_{+}\times\mathbb{R}_{+}^{*}$,

• $\bullet$

$B_{R_{1}}^{c}\subset\omega_{b}$ and $a(x)=\beta b(x)$, $\ |x|\geqslant R_{1}$, for some $\beta>0$, if $\ m_{1}=m_{2}=0$.

For $\gamma\in\mathbb{R}_{+}^{*}$, we set

[TABLE]

and

[TABLE]

With this notation, we can state the stability result for the system $\eqref{pb311}$.

Theorem 1.1**.**

Let $\gamma\in\mathbb{R}_{+}^{*}$ and $(m_{1},m_{2})\in\{(0,0)\}\cup\mathbb{R}_{+}\times\mathbb{R}_{+}^{*}$. We assume that $\omega_{b}$ satisfies the GCC and that the assumptions $(\mathcal{A}_{1})$ and $(\mathcal{A}_{2})$ hold. Then for any solution $(u,v)$ of the system $\eqref{pb311}$ with initial data $(u_{0},u_{1},v_{0},v_{1})\in\mathcal{H}_{\gamma}$, we have

[TABLE]

where $C$ is positive constant. In addition for $(u_{0},u_{1},v_{0},v_{1})\in\mathcal{H}$, $E_{u,v}(t)$ converges to zero as $t$ goes to infinity.

In the case of Klein-Gordon-type systems we obtain the following uniform decay.

Corollary 1**.**

Let $m_{1},m_{2},\gamma\in\mathbb{R}_{+}^{*}$. Assume that $\omega_{b}$ satisfies the GCC and the assumptions $(\mathcal{A}_{1})$ and $(\mathcal{A}_{2})$ hold.

• $\triangleright$

If $\gamma=1$, then there exist positive constants $C$ and $\alpha$ such that

[TABLE]

for all solution $(u,v)$ of the system $\eqref{pb311}$ with initial data $(u_{0},u_{1},v_{0},v_{1})\in\mathcal{H}_{1}$.

• $\triangleright$

If $\gamma\neq 1$, then there exists a positive constant $C$ such that

[TABLE]

for all solution $(u,v)$ of the system $\eqref{pb311}$ with initial data $(u_{0},u_{1},v_{0},v_{1})\in D(\mathcal{A}^{n})$.

Remark 1**.**

• $\bullet$

To our best knowledge, our result is new for the indirect stabilization problem in exterior domains.

• $\bullet$

Remark that, when $\gamma=1$, the energy of the system $\eqref{pb311}$ decays as fast as that of the corresponding scalar damped equation. So the coupling through velocities, in this case, allows a full transmission of the damping effects, quite different from the coupling through the displacements.

• $\bullet$

To prove our main result we study the energy first at infinity ( Section 2) and then in bounded regions (Section 3). Keeping, only the second step, we can obtain the expnential energy decay for the system $\eqref{pb311}$ in bounded domains with Dirichly boundary condition.

• $\bullet$

Due to technical difficulties we did not cover the Klein-Gordon-Wave case ($m_{1}>0$, $m_{2}=0$); we will be interested in the forthcoming work.

We conclude this introduction with an outline of the rest of this paper. In Section 2 we estimate the total energy at infinity by multiplier arguments. Section 3 is devoted to the study of the energy in bounded domain. The proof of this result is based on observability estimate for scalar wave equation. In order to control the compact terms, we prove in section 4 a weak observability estimate that is based on a unique continuation result. Finally, in Section 5 we combine the results of the previous sections to established our main results.

We denote by $\Omega_{R}:=\Omega\cap B_{R}$ , $C_{R,R^{\prime}}=\Omega\cap(B_{R^{\prime}}\backslash B_{R}),$ when $0<R<R^{\prime}$ ,

[TABLE]

[TABLE]

and $A\lesssim B$ means $A\leqslant CB$ for some positive constante $C$.

2. Estimate of energy near infinity

The main result of this section is as follows.

Proposition 2.1**.**

Let $\gamma\in\mathbb{R}_{+}^{*}$ and $(m_{1},m_{2})\in\{(0,0)\}\cup\mathbb{R}_{+}\times\mathbb{R}_{+}^{*}$. Let $R_{1}>0$ be such that $(\mathcal{A}_{2})$ is satisfied and $R_{2}>R_{1}$. Then for every $\varepsilon>0$, there exists $C_{\varepsilon}>0$ such that for all solution $(u,v)$ of $\eqref{pb311}$ with initial data $(u_{0},u_{1},v_{0},v_{1})\in\mathcal{H}_{\gamma}$, we have

[TABLE]

[TABLE]

for all $t>0$.

Let $\varphi\in C^{\infty}(\mathbb{R}^{d})$ be a function satisfying $0\leqslant\varphi\leqslant 1$ and

[TABLE]

To prove Proposition 2.1, we need the following Lemma.

Lemma 2.1**.**

We assume the hypothesis of Proposition 2.1 and we consider $\varphi$ as above. Then for every $\varepsilon>0$, there exist $C_{\varepsilon}>0$ such that for all solution $(u,v)$ of $\eqref{pb311}$ with initial data $(u_{0},u_{1},v_{0},v_{1})\in\mathcal{H}_{\gamma}$, we have

[TABLE]

[TABLE]

for all $t>0$.

Proof of Lemma 2.1.

Multiplying the first and the second equation of $\eqref{pb311}$ respectively by $\varphi\partial_{t}v$ and $\frac{1}{\gamma^{2}}\varphi\partial_{t}u$ and integrating the sum of these results on $[0,t]\times\Omega$, we obtain

[TABLE]

Note that

[TABLE]

[TABLE]

Then using Young’s inequality, we get

[TABLE]

where

[TABLE]

By hypothesis

[TABLE]

so, we deduce that

[TABLE]

[TABLE]

Using the energy decay $\eqref{decroissance}$ and the fact that $(m_{1},m_{2})\in\{(0,0)\}\cup\mathbb{R}_{+}\times\mathbb{R}_{+}^{*}$, we can see that

[TABLE]

Combining $\eqref{decroissance}$, $\eqref{*0}$ and $\eqref{1}$, we obtain $\eqref{Lemg}$.

∎

Lemma 2.2**.**

Let $\gamma\in\mathbb{R}_{+}^{*}$ and $(m_{1},m_{2})=(0,0)$. Let $R_{1}>0$ be such that $(\mathcal{A}_{2})$ is satisfied and $R_{2}>R_{1}$. Then for every $\varepsilon>0$, there exists $C_{\varepsilon}>0$ such that for all solution $(u,v)$ of $\eqref{pb311}$ with initial data $(u_{0},u_{1},v_{0},v_{1})\in\mathcal{H}_{\gamma}$, we have

[TABLE]

[TABLE]

for all $t>0$.  Where $E^{R_{2}}(v,t)=\frac{1}{2}\int_{|x|>R_{2}}|\partial_{t}v(t,x)|^{2}+|\nabla v(t,x)|^{2}\ dx.$

Proof of Lemma 2.2.

We write the system $\eqref{pb311}$ in the form

[TABLE]

Multiplying the first equation of $\eqref{sysex11}$ by $\varphi v$ and the second one by $\frac{1}{\gamma^{2}}\varphi u$ and integrating the sum of these results on $[0,t]\times\Omega$, we obtain

[TABLE]

where

[TABLE]

According to Lemma 2.1, hypothesis $(\mathcal{A}_{2})$ and using Young’s inequality, we deduce that

[TABLE]

[TABLE]

But we have

[TABLE]

So, for $\varepsilon_{1}$ small enough we get

[TABLE]

[TABLE]

Since

[TABLE]

we deduce that

[TABLE]

Combining this estimate with $\eqref{vgamma}$, we conclude $\eqref{envv}$. This finishes the proof of Lemma 2.2. ∎

Now we give the proof of Proposition 2.1.

Proof of Proposition 2.1.

We distinguish the case $m_{1}=m_{2}=0$ and the case where $m_{1}\in\mathbb{R}_{+}$ and $m_{2}\in\mathbb{R}_{+}^{*}$.

First case $m_{1}=m_{2}=0$. Multiplying the first equation of $\eqref{pb311}$ by $\varphi u$ and integrating on $[0,t]\times\Omega$, we obtain

[TABLE]

[TABLE]

Note that we have

[TABLE]

[TABLE]

So, combining this identity with $\eqref{ugamma}$ and using $\eqref{2.5}$, we get

[TABLE]

[TABLE]

Using that,

[TABLE]

we obtain

[TABLE]

[TABLE]

According to $\eqref{vgamma}$ and using $\eqref{varphi}$, we get

[TABLE]

[TABLE]

where $E^{R_{2}}(u,t)=\frac{1}{2}\int_{|x|>R_{2}}|\partial_{t}u(t,x)|^{2}+|\nabla u(t,x)|^{2}\ dx$.

Combining $\eqref{envv}$ and $\eqref{*22**}$, we conclude $\eqref{enex1g}$.

Second case $m_{1}\in\mathbb{R}_{+}$ and $m_{2}\in\mathbb{R}_{+}^{*}$. Multiplying the first and the second equation of $\eqref{pb311}$ respectively by $\varphi u$ and $\varphi v$ and integrating the sum of these results on $[0,t]\times\Omega$, we obtain

[TABLE]

[TABLE]

Using the following estimates for $\varepsilon_{2}$ small enough

[TABLE]

and according to Lemma 2.1, we infer $\eqref{enex1g}$. The proof of proposition 2.1 is now completed. ∎

3. Estimate of energy in bounded region

In this section, we will study the energy in bounded domain. For this aim, we consider a function $\psi\in C_{0}^{\infty}(\mathbb{R}^{d})$ such that $0\leqslant\psi\leqslant 1$ and

[TABLE]

where $R_{4}>R_{3}>R_{1}$ and $R_{1}>0$ be such that $(\mathcal{A}_{2})$ is satisfied.

It is easy to verify that $(u^{i},v^{i})=(\psi u,\psi v)$ satisfies the following system

[TABLE]

Proposition 3.1**.**

Let $\gamma\in\mathbb{R}_{+}^{*}$, $(m_{1},m_{2})\in\{(0,0)\}\cup\mathbb{R}_{+}\times\mathbb{R}_{+}^{*}$ and $\psi$ be as above. Assume that the assumption $(\mathcal{A}_{1})$ holds and that $(\omega_{b},T)$ geometrically controls $\Omega$ for some $T>0$. Then for every $\varepsilon>0$, there exist $C_{\varepsilon}>0$ such that for all solution $(u,v)$ of $\eqref{pb311}$ with initial data $(u_{0},u_{1},v_{0},v_{1})\in\mathcal{H}_{\gamma}$, we have

[TABLE]

[TABLE]

for all $t>0$. Where

[TABLE]

In order to prove proposition 3.1 we need the following result.

Lemma 3.1**.**

Assume that the hypothesis of Proposition 3.1 hold. Then for every $\varepsilon>0$, there exists $C_{\varepsilon}>0$ such that for all solution $(u,v)$ of $\eqref{pb311}$ with initial data $(u_{0},u_{1},v_{0},v_{1})\in\mathcal{H}_{\gamma}$, we have

[TABLE]

[TABLE]

for all $t>0$.

proof of Lemma 3.1 .

We multiply the first and the second equation of $\eqref{sysin1}$ respectively by $b(x)\partial_{t}v^{i}$ and $\frac{b(x)}{\gamma^{2}}\partial_{t}u^{i}$ and we integrate the sum of these results on $[t,t+T]\times\Omega$, we get

[TABLE]

From Young’s inequality and using hypothesis ($\mathcal{A}_{1}$), we infer that

[TABLE]

[TABLE]

This implies $\eqref{Lem6}$.

∎

Proof of proposition 3.1.

First, we recall the following observability estimate for the wave equation ( see proposition $3$, [11]).

Lemma 3.2**.**

Let $\gamma,T>0$ and $\mathcal{O}$ a bounded domain. Let $\phi$ be a nonnegative function on $\mathcal{O}$ and setting

[TABLE]

We assume that $(\mathcal{V},T)$ satisifies the GCC. There exists $C_{T}>0$, such that for all $(u_{0},u_{1})\in H_{0}^{1}(\mathcal{O})\times L^{2}(\mathcal{O}),f\in L_{loc}^{2}(\mathbb{R}_{+},L^{2}(\mathcal{O}))$, and all $t>0$ the solution of

[TABLE]

where $m\geqslant 0$, satisfies with

[TABLE]

the inequality

[TABLE]

Let $\omega_{b,1}=\omega_{b}\cap B_{R_{4}}=\{x\in\Omega\cap B_{R_{4}},\ b(x)>b^{-}>0\}$. Since $(\omega_{b},T)$ satisfies the ** GCC**, $B_{R_{1}^{c}}\subset\omega_{b}$ and $R_{4}>R_{1}$, we conclude that $(\omega_{b,1},T)$ geometrically controls $\Omega_{R_{4}}$.

So, according to Lemma 3.2 and using hypothesis $(\mathcal{A}_{1})$, we have

[TABLE]

[TABLE]

where

[TABLE]

We have also

[TABLE]

[TABLE]

where

[TABLE]

Adding the two estimates above and using $\eqref{Lem6}$, we deduce that

[TABLE]

[TABLE]

Since $\psi\equiv 1$ for $|x|\leqslant R_{3}$, we get

[TABLE]

Combining this estimate with $\eqref{**}$, we conclude $\eqref{eningg}$. ∎

4. Weak observability estimate

In this section, we prove the following proposition.

Proposition 4.1**.**

Let $\gamma\in\mathbb{R}_{+}^{*}$ and $m_{1},m_{2}\in\mathbb{R}_{+}$. Let $R_{1}>0$ be such that $(\mathcal{A}_{2})$ is satisfied and $R_{5}>R_{1}$. We assume that the assumption $(\mathcal{A}_{1})$ holds. Then for every $T>T_{\omega_{b}}$ and $\alpha>0$, there exists $C_{T,\alpha}>0$, such that for all $(u_{0},u_{1},v_{0},v_{1})\in(H_{0}^{1}(\Omega)\times L^{2}(\Omega))^{2}$, and all $t>0$, the solution of the system $\eqref{pb311}$ satisfies the following inequality

[TABLE]

Proof of Proposition 4.1.

We note that for each $(u_{0},u_{1},v_{0},v_{1})\in(H_{1}^{0}(\Omega)\times L^{2}(\Omega))^{2}$, the solution $(u,v)$ are given as the limit of smooth solutions $(u_{n},v_{n})(t)$ with $(u_{n},v_{n})(0)=(u_{n,0},v_{n,0})\in(C_{0}^{\infty}(\Omega))^{2}$ and $(\partial_{t}u_{n},\partial_{t}v_{n})(0)=(u_{n,1},v_{n,t})\in(C_{0}^{\infty}(\Omega))^{2}$ such that $(u_{n,0},v_{n,0})\to(u_{0},v_{0})\in(H_{0}^{1}(\Omega))^{2}$ and $(u_{n,1},v_{n,1})\to(u_{1},v_{1})\in(L^{2}(\Omega))^{2}$. Note that

[TABLE]

uniformly on the each closed interval $[0,T]$ for any $T>0$. Therefore we may assume that $(u,v)$ is smooth.

To prove the estimate $\eqref{4.9}$, we argue by contradiction. We assume that there exist a positive sequence $(t_{n})$ and a sequence

[TABLE]

of solution of the system $\eqref{pb311}$ with initial data $(u_{n,0},u_{n,1},v_{n,0},v_{n,1})\in(H_{0}^{1}(\Omega)\times L^{2}(\Omega))^{2}$, such that

[TABLE]

Set

[TABLE]

and

[TABLE]

We infer that

[TABLE]

Therefore

[TABLE]

with respect to the weak topology. By Rellich’s lemma, we can assume that

[TABLE]

It is easy to see that the limit $(y,z)$ satisfies the system

[TABLE]

and

[TABLE]

It is clear that $(\partial_{t}y,\partial_{t}z)$ satisfies the following system

[TABLE]

From the first and previous equations in (4.7), we deduce that $b(x)\partial_{t}^{2}z=0$ on $supp(a)$. But $supp(b)\subset supp(a)$, so $\partial_{t}^{2}z=0$ on $supp(b)$. Setting $w=\partial_{t}z$, we have

[TABLE]

Using the first and second equations in (4.8), we can see that $WF^{1}(w)\cap(0,T)\times\omega_{b}\times\mathbb{R}\times\mathbb{R}^{n}$ is a subset of

[TABLE]

where $WF^{1}(w)$ denotes the $H^{1}$-wavefront set of $w$. Since $B_{R_{1}}^{c}\subset\omega_{b}$, we deduce that $w\in H_{loc}^{1}((0,T)\times B_{R_{1}}^{c}).$ Next, we will show that $w\in H_{loc}^{1}([0,T]\times R_{R_{1}})$. Let $\rho_{0}=(t_{0},x_{0},\tau_{0},\xi_{0})\in T^{*}([0,T]\times B_{R_{1}})$ and $\Gamma_{0}$ be the generalized bicharacteristic issued from $\rho_{0}$. Set $\{\rho_{1}:=(0,x_{1},\tau_{1},\xi_{1})\}=\Gamma_{0}\cap\{t=0\}$ and $\{\rho_{2}:=(T,x_{2},\tau_{2},\rho_{2})\}=\Gamma_{0}\cap\{t=T\}$, so we distinguish two cases,

$1^{st}$** case: ** $x_{1}$ or $x_{2}\notin B_{R_{1}}$. In this case $\rho_{1}$ or $\rho_{2}\notin WF^{1}(w)$). Since $T>T_{\omega_{b}}$, then using the propagation of regularity along the bicharacteristic flow of the operator $\partial_{t}^{2}-\gamma^{2}\Delta$ (see [17, 18]), we obtain $\rho_{0}\notin WF^{1}(w)$.

$2^{nd}$** case: ** $x_{1},x_{2}\in B_{R_{1}}$. Since $\rho_{1},\rho_{2}\in T^{*}([0,T]\times B_{R_{1}})$ and $\omega_{b}$ controls geometrically $[0,T]\times\Omega$, then $\Gamma_{0}$ intersects the region $[0,T]\times(\omega_{b}\cap\Omega_{R_{1}})$. But $w\in H_{loc}^{1}([0,T]\times(\omega_{b}\cap\Omega_{R_{1}}))$, then applying again the regularity propagation theorem, we deduce that $\rho_{0}\notin WF^{1}(w)$. Therefore, we conclude that $w\in H_{loc}^{1}((0,T)\times\Omega)$. Now, set $\tilde{w}=\partial_{t}w$. Since $\mathbb{R}^{n}\setminus\Omega_{R_{5}}\subset\omega_{b}$, so $\tilde{w}=0$ on $\mathbb{R}^{n}\setminus\Omega_{R_{5}}$ and satisfies

[TABLE]

Since $\omega_{b}\cap\Omega_{R_{5}}$ controls geometrically $\Omega_{R_{5}}$, then using the classical unique continuation result (see [7, 8] ), we infer that $\tilde{w}\equiv 0$ on $(0,T)\times\Omega_{R_{5}}$. Therefore, the function $z$ satisfies

[TABLE]

This implies that $z=0$ on $(0,T)\times\Omega$. Now, from $\eqref{pbg}$ we obtain

[TABLE]

Arguing as for $z$, we can prove that $y=0$. This is in contradiction with $\eqref{0*}$.

∎

5. Proof of Theorem 1.1

Let $R_{2}>R_{1}$. According to $\eqref{enex1g}$ for $t=nT$, $n\in\mathbb{N}^{*}$, we have

[TABLE]

[TABLE]

Next, using $\eqref{eningg}$ with $R_{3}=2R_{2}$ and $R_{4}=3R_{2}$, we get

[TABLE]

[TABLE]

Thus

[TABLE]

[TABLE]

This gives

[TABLE]

[TABLE]

From the following estimate

[TABLE]

and using $\eqref{decroissance}$ and $\eqref{4.6}$, we deduce that

[TABLE]

[TABLE]

So, combining $\eqref{4.5}$ and $\eqref{4.6}$, we conclude for small enough $\varepsilon$ the following estimate

[TABLE]

[TABLE]

Next, From $\eqref{4.9}$ with $R_{5}=3R_{2}$ we have

[TABLE]

Thus

[TABLE]

Finally, using $\eqref{4.10}$ for $\alpha$ small enough in $\eqref{enn}$, we find

[TABLE]

Therefore

[TABLE]

As the energy is decreasing then

[TABLE]

[TABLE]

On the other hand, using $\eqref{enex1g}$, $\eqref{4.10}$ and $\eqref{4.11}$, we deduce that

[TABLE]

Since $\varphi\equiv 1$ for $|x|\geqslant R_{2}$,

[TABLE]

therefore

[TABLE]

Poincare’s inequality and the fact that the energie of $(u,v)$ is decreasing gives

[TABLE]

for all $t>0$.

Adding $\eqref{4.14}$ and $\eqref{4.13}$, we infer that

[TABLE]

for all $t>0$.

Proof of Corollary 1.

From $\eqref{4.12}$, we deduce if $\gamma=1$

[TABLE]

we choose $t$ such that $\frac{C}{t}<1$ and using the semi-group proprety, we conclude that the estimate $\eqref{theorem 2}$.

and if $\gamma\neq 1$,

[TABLE]

according to [Theoreme $2.1$, References] we infer that $\eqref{theorem 22}$.

∎