Behaviors of the energy of solutions of two coupled wave equations with nonlinear damping on a compact manifold with boundary

TL;DR

This paper investigates how the energy of solutions to coupled wave equations with nonlinear damping on a compact manifold decays over time, revealing that decay rates are governed by a specific differential equation.

Contribution

It establishes decay rate estimates for coupled wave equations with nonlinear damping, under geometric conditions, extending understanding of indirect damping effects on energy decay.

Findings

Energy decay rates are characterized by a first order differential equation.

Decay behavior depends on geometric conditions of coupling and damping regions.

Results apply to smooth solutions on compact manifolds with boundary.

Abstract

In this paper we study the behaviors of the the energy of solutions of coupled wave equations on a compact manifold with boundary in the case of indirect nonlinear damping . Only one of the two equations is directly damped by a localized nonlinear damping term. Under geometric conditions on both the coupling and the damping regions we prove that the rate of decay of the energy of smooth solutions of the system is determined from a first order differential equation .

Equations

$$\left\{\begin{array}[]{ll}\partial_{t}^{2}u-\Delta u+b(x)v+a(x)g(\partial_{t}u)=0&\text{in }\mathbb{R}_{+}^{\ast}\times\Omega\\ \partial_{t}^{2}v-\Delta v+b(x)u=0&\text{in }\mathbb{R}_{+}^{\ast}\times\Omega\\ u=v=0&\text{on }\mathbb{R}_{+}^{\ast}\times\Gamma\\ \left(u\left(0,x\right),\partial_{t}u\left(0,x\right)\right)=\left(u_{0},u_{1}\right)\text{ and }\left(v\left(0,x\right),\partial_{t}v\left(0,x\right)\right)=\left(v_{0},v_{1}\right)&\text{in }\Omega,\end{array}\right.$$

$$\begin{array}[]{l}g\left(y\right)y\leq M_{0}y^{2},~{}~{}\left|y\right|<1\\ my^{2}\leq g\left(y\right)y\leq My^{2},~{}~{}\left|y\right|\geq 1\\ \left\|g^{\prime}\right\|_{L^{\infty}}\leq M_{1},\end{array}$$

$$\begin{array}[]{l}E_{u,v}\left(t\right)=\frac{1}{2}\mathop{\displaystyle\int}_{\Omega}\left|\nabla u\left(t,x\right)\right|^{2}+\left|\nabla v\left(t,x\right)\right|^{2}+\left|\partial_{t}u\left(t,x\right)\right|^{2}+\left|\partial_{t}v\left(t,x\right)\right|^{2}dx\\ \text{ \ \ \ \ \ \ \ \ \ \ }+\mathop{\displaystyle\int}_{\Omega}b\left(x\right)u\left(t,x\right)v\left(t,x\right)dx.\end{array}$$

$${\Vert b\Vert }_{\infty }\le \frac{1-\delta }{{\lambda }^2},$$

$$\begin{array}[]{l}E_{u,v}\left(0\right)=\frac{1}{2}\mathop{\displaystyle\int}_{\Omega}\left|\nabla u_{0}\left(x\right)\right|^{2}+\left|\nabla v_{0}\left(x\right)\right|^{2}+\left|u_{1}\left(x\right)\right|^{2}+\left|v_{1}\left(x\right)\right|^{2}dx+\mathop{\displaystyle\int}_{\Omega}b\left(x\right)u_{0}\left(x\right)v_{0}\left(x\right)dx\\ \text{ \ \ \ \ \ \ \ \ \ \ }\geq\frac{\delta}{2}\mathop{\displaystyle\int}_{\Omega}\left|\nabla u_{0}\left(x\right)\right|^{2}+\left|\nabla v_{0}\left(x\right)\right|^{2}+\left|u_{1}\left(x\right)\right|^{2}+\left|v_{1}\left(x\right)\right|^{2}dx,\end{array}$$

$$\left\{\begin{array}[]{c}\dfrac{d}{dt}U+\mathcal{A}U+\mathcal{B}U=0\\ U\left(0\right)=U_{0}\in\mathcal{H}\end{array}\right.$$

$$\begin{array}[]{c}U=\left(\begin{array}[]{c}u\\ v\\ \partial_{t}u\\ \partial_{t}v\end{array}\right),U_{0}=\left(\begin{array}[]{c}u_{0}\\ v_{0}\\ u_{1}\\ v_{1}\end{array}\right),\end{array}$$

$$\begin{array}[]{c}\mathcal{A=}\left(\begin{array}[]{cccc}0&0&-Id&0\\ 0&0&0&-Id\\ -\Delta&b&0&0\\ b&-\Delta&0&0\end{array}\right)\end{array}$$

$$\begin{array}[]{l}D\left(\mathcal{A}\right)=\left\{U\in\mathcal{H};\mathcal{A}U\in\mathcal{H}\right\}\\ =\left(H_{0}^{1}\left(\Omega\right)\cap H^{2}\left(\Omega\right)\right)\times\left(H_{0}^{1}\left(\Omega\right)\cap H^{2}\left(\Omega\right)\right)\times H_{0}^{1}\left(\Omega\right)\times H_{0}^{1}\left(\Omega\right),\end{array}$$

$$\begin{array}[]{c}\mathcal{B}U=\left(\begin{array}[]{c}0\\ 0\\ a\left(x\right)g\left(\partial_{t}u\right)\\ 0\end{array}\right)\end{array}$$

$$\begin{array}[]{c}E_{u,v}\left(t\right)-E_{u,v}\left(0\right)=-\mathop{\displaystyle\int}_{0}^{t}\mathop{\displaystyle\int}_{\Omega}a\left(x\right)g\left(\partial_{t}u\left(s,x\right)\right)\partial_{t}u\left(s,x\right)dxds\end{array}$$

$$\begin{array}[]{c}E_{\partial_{t}u,\partial_{t}v}\left(t\right)-E_{\partial_{t}u,\partial_{t}v}\left(0\right)=-\mathop{\displaystyle\int}_{0}^{t}\mathop{\displaystyle\int}_{\Omega}a\left(x\right)g^{\prime}\left(\partial_{t}u\left(s,x\right)\right)\left|\partial_{t}^{2}u\left(s,x\right)\right|^{2}dxds.\end{array}$$

$$\left\{\begin{array}[]{ll}\partial_{t}^{2}u_{1}-\Delta u_{1}+b(x)u_{2}=0&\text{in }\left(0,T_{0}\right)\times\Omega\\ \partial_{t}^{2}u_{2}-\Delta u_{2}+b(x)u_{1}=0&\text{in }\left(0,T_{0}\right)\times\Omega\\ u_{1}=u_{2}=0&\text{on }\left(0,T_{0}\right)\times\Gamma\\ a\left(x\right)u_{1}=0&\text{on }\left(0,T_{0}\right)\times\Omega\\ u_{1}\in H^{1}\left(\left(0,T_{0}\right)\times\Omega\right)\text{ and }u_{2}\in L^{2}\left(\left(0,T_{0}\right)\times\Omega\right),&\end{array}\right.$$

$$h_0\left(g\left(s\right)s\right)\ge {ϵ}_0\left({\left|s\right|}^2+{\left|g\left(s\right)\right|}^2\right),\left|s\right|\le 1,$$

$$\begin{array}[]{c}h\left(s\right)=m_{a}\left(\Omega\right)h_{0}\left(\frac{s}{m_{a}\left(\Omega\right)}\right),\text{ where }m_{a}=a\left(x\right)dx.\end{array}$$

$$\begin{array}[]{c}\underset{s\rightarrow 0}{\lim}h^{-1}\left(s\right)=\underset{s\rightarrow 0}{\lim}\left(h^{-1}\right)^{\prime}\left(s\right)=\underset{s\rightarrow 0}{\lim}s\left(h^{-1}\right)^{\prime\prime}\left(s\right)=\underset{s\rightarrow 0}{\lim}s^{2}\left(h^{-1}\right)^{\prime\prime\prime}\left(s\right)=0,\end{array}$$

$$\begin{array}[]{l}\underset{s\rightarrow\infty}{\lim}s\left(h^{-1}\right)^{\prime}\left(1/s^{\beta}\right)=\alpha_{0},\\ \left(h^{-1}\right)^{\prime}\left(s\right)\leq\beta s\left(h^{-1}\right)^{\prime\prime}\left(s\right),\text{ for all }s\in\left[0,r_{0}\right],\\ \left(\beta^{2}-\beta\right)s\left(h^{-1}\right)^{\prime\prime}\left(s\right)+\beta^{2}s^{2}\left(h^{-1}\right)^{\prime\prime\prime}\left(s\right)\geq 0,\text{ for all }s\in\left[0,r_{0}\right].\end{array}$$

$$\begin{array}[]{c}\frac{\left(h^{-1}\right)^{\prime}\left(s\right)\left(\left(\beta^{2}-\beta\right)s\left(h^{-1}\right)^{\prime\prime}\left(s\right)+\beta^{2}s^{2}\left(h^{-1}\right)^{\prime\prime\prime}\left(s\right)\right)}{\beta s\left(h^{-1}\right)^{\prime\prime}\left(s\right)-\left(h^{-1}\right)^{\prime}\left(s\right)}\leq\alpha_{1},\text{ for all }s\in\left[0,r_{0}\right].\end{array}$$

$$\begin{array}[]{c}\vskip 6.0pt plus 2.0pt minus 2.0ptE_{u,v}\left(t\right)\leq\frac{c}{t}\sum_{i=0}^{1}E_{\partial_{t}^{i}u,\partial_{t}^{i}v}\left(0\right),~{}t>0,\\ \vskip 6.0pt plus 2.0pt minus 2.0ptU_{0}=\left(u_{0},v_{0},u_{1},v_{1}\right)\in\left(H_{0}^{1}\left(\Omega\right)\cap H^{2}\left(\Omega\right)\right)^{2}\times\left(H_{0}^{1}\left(\Omega\right)\right)^{2},\end{array}$$

$$\begin{array}[]{l}\vskip 6.0pt plus 2.0pt minus 2.0ptE_{u,v}\left(t\right)\leq C\left(1+\sum_{i=0}^{1}E_{\partial_{t}^{i}u,\partial_{t}^{i}v}\left(0\right)\right)\left(\varphi\left(t\right)\right)^{-1},~{}t>0,\\ \vskip 6.0pt plus 2.0pt minus 2.0ptU_{0}=\left(u_{0},v_{0},u_{1},v_{1}\right)\in\left(H_{0}^{1}\left(\Omega\right)\cap H^{2}\left(\Omega\right)\right)^{2}\times\left(H_{0}^{1}\left(\Omega\right)\right)^{2},\end{array}$$

$$\begin{array}[]{l}\dfrac{d\varphi}{dt}-\frac{\epsilon_{0}}{2C_{1}}\varphi\left(h^{-1}\right)^{\prime}\left(1/\varphi^{\beta}\right)=0,~{}0<\varphi^{-\beta}\left(0\right)\leq r_{0}~{}\text{such that}\\ \left(h^{-1}\right)^{\prime}\left(\varphi^{-\beta}\left(0\right)\right)<\inf\left(\frac{2C_{1}\delta}{\epsilon_{0}}\left(8C_{T}+2\lambda^{2}+1\right)^{-1},\frac{1}{\epsilon_{0}}\left(\frac{1}{m}+M^{2}\right)^{-1}\right),\end{array}$$

$$\begin{array}[]{c}\text{for }U_{0}\in\mathcal{H},\underset{t\rightarrow\infty}{\lim}\vskip 6.0pt plus 2.0pt minus 2.0ptE_{u,v}\left(t\right)=0.\end{array}$$

$$\begin{array}[]{c}\left\{\begin{array}[]{l}-\Delta u+b(x)v=0\\ -\Delta v+b(x)u=0\\ \begin{array}[]{ll}u=v=0&\text{on }\Gamma\end{array}\end{array}\right.\end{array}$$

$$\begin{array}[]{c}\dfrac{d\theta}{dt}+\frac{\epsilon_{0}}{2C_{1}}\theta\left(h^{-1}\right)^{\prime}\left(\theta^{\beta}\right)=0.\end{array}$$

$${\left(h^{-1}\right)}^{\prime }\left(s\right)\le \alpha s{\left(h^{-1}\right)}^{\prime \prime }\left(s\right),\text{ for all }s\in (0,r_0].$$

$$\frac{d\theta }{dt}+C\theta {\left(h^{-1}\right)}^{\prime }\left({\theta }^{\beta }\right)=0,\text{such that }0<\theta \left(0\right)\le r_0,$$

$$\begin{array}[]{c}\theta\left(t\right)\leq\left(\left(\left(h^{-1}\right)^{\prime}\right)^{-1}\left(\frac{\alpha/\beta C}{t+k_{0}}\right)\right)^{1/\beta},\text{ for all }t\geq 0,\end{array}$$

$$\frac{\alpha /\beta C}{{\left(h^{-1}\right)}^{\prime }\left(r_0\right)}\le k_0\le \frac{\alpha /\beta C}{{\left(h^{-1}\right)}^{\prime }\left({\theta }^{\beta }\left(0\right)\right)}.$$

$$\psi \left(t\right)={\left({\left({\left(h^{-1}\right)}^{\prime }\right)}^{-1}\left(\frac{\alpha /\beta C}{t+k_0}\right)\right)}^{1/\beta },\text{ for all }t\ge 0.$$

$${\psi }^{\prime }\left(t\right)\ge -\frac{\alpha }{\beta }\frac{\psi \left(t\right)}{t+k_0},\text{ for all }t\ge 0.$$

Full text

1. Introduction and Statement of the results

Let $\left(\Omega,g_{0}\right)$ be a $C^{\infty}$ compact connected n-dimensional Riemannian manifold with boundary $\Gamma.$ We denote by $\Delta$ the Laplace-Beltrami operator on $\Omega$ for the metric $g_{0}.$ We consider a system of coupled wave equations with nonlinear damping

[TABLE]

where $g:\mathbb{R}\longrightarrow\mathbb{R}$ is a continuous, monotone increasing function, $g(0)=0$. In addition we assume that

[TABLE]

for some positive real numbers $M_{0},~{}m,~{}M~{}$ and $M_{1}$. In this paper, we deal with real solutions, the general case can be treated in the same way. With the system above we associate the energy functional given by

[TABLE]

We assume that $a$ and $b$ are two nonnegative smooth functions such that

[TABLE]

for some $\delta>0,$ where $\lambda$ is the Poincaré’s constant on $\Omega.$ Under these assumptions we have

[TABLE]

for all $\left(u_{0},v_{0},u_{1},v_{1}\right)\in\mathcal{H}=~{}H_{0}^{1}\left(\Omega\right)\times H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)\times L^{2}\left(\Omega\right).$

The nonlinear evolution equation $\left(\ref{system intro}\right)$ can be rewritten under the form

[TABLE]

where

[TABLE]

and the unbounded operator $\mathcal{A}$ on $\mathcal{H}$ is defined by

[TABLE]

with domain

[TABLE]

and

[TABLE]

Under our assumptions and from the nonlinear semi-group theory (see for example [5]), we can infer that for $U_{0}\in\mathcal{H},$ the problem $\left(\ref{abstract system}\right)$ admits a unique solution $U\in C^{0}\left(\mathbb{R}_{+},\mathcal{H}\right).$ Moreover we have the following energy estimate

[TABLE]

for all $t\geq 0.$ In addition, since $g^{\prime}\in L^{\infty}\left(\mathbb{R}\right),$ then if $U_{0}\in D\left(\mathcal{A}\right),$ we have $U\in C\left(\mathbb{R}_{+},D\left(\mathcal{A}\right)\right)$ and

[TABLE]

The systems like $\left(\ref{system intro}\right)$ appear in many physical situations. Indirect damping of reversible systems occurs in several applications in engineering and mechanics. In general it is impossible or too expansive to damp all the components of the state, so it is important to study stabilization properties of coupled systems with a reduced number of feedbacks.

The case of a linear damping and constant coupling b in $\left(\ref{system intro}\right)$ has already been treated in [3]. They showed that the System $\left(\ref{system intro}\right)$ cannot be exponentially stable and that the energy decays polynomially. In [2] Alabau et al generalized these results to cases for which the coupling $b=b(x)$ and the damping term $a=a(x)$ satisfy the Piecewise Multipliers Geometric Condition (PMGC) [2]. This geometric assumption is a generalization of the usual multiplier geometric condition (or $\Gamma$-condition) of [Zua, 16] and is much more restrictive than the sharp Geometric Control Condition (GCC). In [1] Alabau et al generalized this result and they proved that the system $\left(\ref{system intro}\right)$ is polynomially stable when the regions $\{a>0\}$ and $\{b>0\}$ both satisfy the Geometric Control Condition and the coupling term satisfies a smallness assumption.  This result has generalized by Aloui et al [7], by assuming a more natural smallness condition on the infinity norm of the coupling term $b.$ Finally we quote the result of Fu [13] in which he shows the logarithmic decay property without any geometric conditions on the effective damping domain.

The problem of the indirect nonlinear damping has been studied by Alabau et Al [5] when the system is coupled by the velocity.  In this case they show that the energy of these kinds of system decays as fast as that of the corresponding scalar nonlinearly damped equation. Hence, the coupling through velocities allows a full transmission of the damping effects. To our knowledge no results seems to be known in the case of indirect nonlinear damping for a coupled system coupled in displacements.

The goal of this paper is to determine the rate of decay of the energy of coupled wave system with indirect nonlinear damping and coupled in displacements. More precisely, we prove, under some geometric conditions on the localized damping domain and the localized coupling domain, that the rate of decay of the energy is determined from a first order differential equation . In addition, we obtain that if the behavior of the damping is close to the linear case, then the linear and the nonlinear case has the same rate of decay. In the other case we find that the rate of decay of the coupled system is close to the one obtained for a single damped wave equation.

The optimality of our results is a open questions. Lower energy estimates have been established in [4, 5] and [6] for scalar one-dimensional wave equations, scalar Petrowsky equations in two-dimensions and one-dimensional wave systems coupled by velocities. These results can be extended to the case of one-dimensional wave systems coupled by displacement. In our case we obtain a quasi-optimal energy decay formula when the behavior of the damping is not close to the linear one.

A natural necessary and sufficient condition to obtain controllability for wave equations is to assume that the control set satisfies the Geometric Control Condition (GCC) defined in [8, 18]. For a subset $\omega$ of $\Omega$ and $T>0$, we shall say that $\left(\omega,T\right)$ satisfies GCC if every geodesic traveling at speed one in $\Omega$ meets $\omega$ in a time $t<T$. We say that $\omega$ satisfies GCC if there exists $T>0$ such that $\left(\omega,T\right)$ satisfies GCC. We also set $T_{\omega}=inf\left\{T>0;(\omega,T)\text{ satisfies }GCC\right\}.$

We denote by $\omega=\left\{a\left(x\right)>0\right\}$ the control set and by $\mathcal{O}=\left\{b\left(x\right)>0\right\}$ the coupling set.

**Assumption $\left(A1\right)$: : **

Unique continuation property:

There exists $T_{0}>0,$ such that the only solution of the system

[TABLE]

is the null one $u_{1}=u_{2}=0$.

Note that the unique continuation assumption above is valid if we assume that $\omega\cap\mathcal{O}$ satisfies the GCC (see [7]). Also according to Alabau et al. [1, Proposition 4.7] we have the following result$:$ We assume that $\omega$ and $\mathcal{O}$ satisfy the GCC, then if $\left\|b\right\|_{\infty}\leq\min\left(\frac{1}{5\lambda^{2}},\frac{1}{50\lambda\sqrt{C_{T_{\omega}}}}\right),$ there exists $T^{\ast}\geq\max\left(T_{\omega},T_{\mathcal{O}}\right)$ such that if $T_{0}>T^{\ast}$ then the only solution of the system $\left(\ref{unique continuation system}\right)$ is the null one.

In order to characterize decay rates for the energy, we need to introduce several special functions, which in turn will depend on the growth of $g$ near the origin. According to [14] there exists a concave continuous, strictly increasing function $h_{0}$, linear at infinity with $h_{0}(0)=0$ such that

[TABLE]

where $\epsilon_{0}$ is a positive constant. We set

[TABLE]

**Assumption $\left(A2\right)$: **

We assume that there exists $0<r_{0}\leq 1$ such that the function $h^{-1}\in C^{3}\left((0,r_{0}]\right)$ and strictly convex. In addition we suppose that

[TABLE]

and there exist $\beta>1$ and $\alpha_{0}>0,$ such that

[TABLE]

Moreover, we assume that if $\beta s\left(h^{-1}\right)^{\prime\prime}\left(s\right)-\left(h^{-1}\right)^{\prime}\left(s\right)>0,$ for all $s\in(0,r_{0}],$ then there exists $\alpha_{1}>0,$ such that

[TABLE]

We know that in the case of linear damping we have

[TABLE]

so we cannot expect to obtain a better rate of decay in the case of nonlinear damping. More precisely we have the following result.

Theorem 1**.**

We suppose that $a$ and $b$ are two smooth non-negative functions and the conditions (1.3) and the assumption A2 hold. In addition, we assume that $\omega$ and $\mathcal{O}$ satisfy the GCC and the assumption A1 holds. The solution $U\left(t\right)=\left(u\left(t\right),v\left(t\right),\partial_{t}u\left(t\right),\partial_{t}v\left(t\right)\right)$ of the system $\left(\ref{system intro}\right)$ then satisfies

[TABLE]

where $C$ is positive constant and $\varphi$ is a solution of the following ODE

[TABLE]

where $C_{1}=C\left(T,\left\|b\right\|_{\infty},\left\|a\right\|_{\infty}\right)$. Moreover,

[TABLE]

Remark 1**.**

The smallness condition on the infinity norm of $b$ is required to ensure that the only solution of the system

[TABLE]

is the null one.

Remark 2**.**

We note that $\theta\left(t\right)=\dfrac{1}{\varphi\left(t\right)}$ is a solution of the following ODE

[TABLE]

To prove our result it is sufficient to show the integrability of $\varphi^{\prime}E_{u,v}$ on $(0,\infty)$. For this purpose we show an estimate on a functional $X(t)$ which is equivalent to the weighted energy functional (See [11] for similar idea). Also we prove a weighted observability estimate for the wave equation with a potential. In addition, we use the unique continuation hypotheses (A1) to prove a weak observability estimate of the weighted $L^{2}$-norm of the solution.

1.1. Some examples of decay rates and lower energy estimates

We give some examples of feedback growths together with the resulting energy decay rate when applying our results. For clarity of exposition we will deal with the damping which satisfies strict bounds, i.e. by saying $g(s)s\simeq f(s)$ we will mean there are constants $m,M$ so that $mf(s)\leq g(s)s\leq Mf(s)$. In the sequel $\tilde{C}$ denotes a generic positive constant which is independents of the energy of the initial data and setting $E_{0}=1+\sum_{i=0}^{1}E_{\partial_{t}^{i}u,\partial_{t}^{i}v}\left(0\right)$. Below we assume that $\varphi\left(0\right)$ verifies the condition $\left(\ref{ode theorem 1}\right).$

First we give an explicit upper bound of solutions of the ordinary differential equation $\left(\ref{one over phi ode}\right).$

Lemma 1**.**

We assume that there exists $0<r_{0}\leq 1$ such that the function $h^{-1}\in C^{2}\left((0,r_{0}]\right),$ monotone increasing and strictly convex. In addition we suppose that there exists $\alpha>0,$ such that

[TABLE]

Let $\theta$ be a solution of the following ODE

[TABLE]

where $C$ is a positive constant and $\beta\geq 1$. We have

[TABLE]

where

[TABLE]

Proof.

Let

[TABLE]

Direct computations and $\left(\ref{assumption2}\right)$, give

[TABLE]

On the other hand, it is easy to see that

[TABLE]

Therefore, using $\left(\ref{k assumption}\right)$, we conclude that

[TABLE]

The desired result follows from [10, Lemma 1].

Now we give some examples.

Example 1** (Linearly bounded case).**

Suppose $g\left(s\right)s\simeq s^{2}$. According to (1.9), auxiliary function $h_{0}$ which may be defined as $h_{0}(y)=(cy)^{\gamma}$ with $1/2<\gamma<1$ and for suitable constant $c>0$. We use the ODE

[TABLE]

Consequently,

[TABLE]

Example 2**.**

Suppose $g\left(s\right)s\simeq s^{2}\left(\ln\left(1/s\right)\right)^{-p}$, $0<\left|s\right|<1/2$ for some $p>0$. According to (1.9), auxiliary function $h_{0}$ which may be defined as $h_{0}(y)=(cy)^{\gamma}$ with $1/2<\gamma<1$ and for suitable constant $c>0$. We use the ODE

[TABLE]

Consequently,

[TABLE]

Example 3** (The Polynomial Case).**

Suppose $g\left(s\right)s\simeq|s|^{p+1}$, $0<\left|s\right|<1$ for some $p>1$. According to (1.9), auxiliary function $h_{0}$ which may be defined as $h_{0}(y)=(cy)^{2/(p+1)}$ for suitable constant $c>0$ (determined by the coefficients in the polynomial bound on the damping $g(s)$). We use the ODE

[TABLE]

Consequently,

[TABLE]

where

[TABLE]

Example 4** (Exponential damping at the origin).**

Assume: $g\left(s\right)=s^{3}e^{-1/s^{2}}$, $0<\left|s\right|<1$. First we need to determine $h_{0}$ according to (1.9). Setting $h_{0}(g(y)y)=cy^{2}$, we see that

[TABLE]

We use the ODE

[TABLE]

to obtain

[TABLE]

for all $\beta>1.$

Example 5** (Exponential damping at the origin).**

Assume: $g\left(s\right)=s^{3}e^{-e^{1/s^{2}}}$, $0<\left|s\right|<1$. First we need to determine $h_{0}$ according to (1.9). Setting $h_{0}(g(y)y)=cy^{2}$, we see that

[TABLE]

We use the ODE

[TABLE]

to obtain

[TABLE]

for all $\beta>1.$

We finish this part by giving a result on the lower estimate of the energy of the one-dimensional coupled wave system.

Proposition 1**.**

We suppose that $\Omega=(0,1)$ and $g$ is a odd function. We set

[TABLE]

We assume that $a$ and $b$ are two smooth non-negative functions and the conditions (1.3) and the assumption A2 hold. In addition, we suppose that $\omega$ and $\mathcal{O}$ satisfy the GCC and the assumption A1 holds. Let $U\left(t\right)=\left(u\left(t\right),v\left(t\right),\partial_{t}u\left(t\right),\partial_{t}v\left(t\right)\right)$ be the solution of the system $\left(\ref{system intro}\right),$ then there exists $T_{0}>0$ such that

[TABLE]

where $\psi$ is a solution of the following ODE

[TABLE]

Proof.

We proceed as in [5] and using the fact that

[TABLE]

we see that there exists $T_{0}>0$ such that

[TABLE]

Since $\psi$ is a solution of $\left(\ref{psi lower ode}\right),$ then using [10, Lemma 1] we conclude that

[TABLE]

2. Proof of Theorem 1

First we give the following weighted observability estimate for the wave equation with potential.

Proposition 2**.**

Let $\gamma,\delta>0$ and $\chi\in L^{\infty}(\Omega)$ satisfying

• •

$\chi\geq 0$* or else,*

• •

$\left\|\chi\right\|_{\infty}\leq\frac{\gamma^{2}-\delta}{\lambda^{2}}$.

Let $\phi$ be a positive function in $C^{2}\left(\mathbb{R}_{+}\right)$ such that

[TABLE]

In addition, if $\phi^{\prime}$ is not the null function, we assume that  there exists a positive constant $K$ such that

[TABLE]

Moreover we suppose that the function

[TABLE]

We consider also $\psi$ nonnegative smooth function on $\Omega$ such that the set $\left(\mathcal{V}:=\left\{\psi\left(x\right)>0\right\},T\right)$ satisfies the GCC. Then there exists $C_{T}>0,$ such that for all $\left(u_{0},u_{1}\right)\in H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right),$ $f\in L_{loc}^{2}\left(\mathbb{R}_{+},L^{2}\left(\Omega\right)\right)$ and all $t>0,$ the solution of

[TABLE]

satisfies with

[TABLE]

the inequality

[TABLE]

Proof.

First we remark that

[TABLE]

To prove the estimate $\left(\ref{observability by a}\right),$ we argue by contradiction. We assume that there exist a positive sequence $\left(t_{n}\right),~{}$a sequence of functions $f_{n}\in L_{loc}^{2}\left(\mathbb{R}_{+},L^{2}\left(\Omega\right)\right)$ and a sequence $\left(u_{n}\left(t\right)\right)$ of solutions of the system $\left(\ref{System onservability lemma}\right)$ with initial data $\left(u_{0,n},u_{1,n}\right)\in H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right),$ such that

[TABLE]

**1st case: **

$t_{n}\underset{n\rightarrow+\infty}{\longrightarrow}\infty.$ Setting

[TABLE]

Therefore $v_{n}$ is a solution of the following system

[TABLE]

where

[TABLE]

Thanks to $\left(\ref{proof lemma 1 observa bility wave contradiction}\right),$ we get

[TABLE]

Using Poincare’s inequality we deduce that

[TABLE]

Utilizing the first part of $\left(\ref{lemm cotrad}\right)$ and the estimate above, we deduce that there exists $~{}\alpha_{1}>0,$ such that

[TABLE]

A combination of the first part of $\left(\ref{lemm cotrad}\right),$ $\left(\ref{PHY assumption 1}\right)$ and $\left(\ref{PHY assumption 2}\right),$ gives

[TABLE]

It is easy to see that

[TABLE]

noting that in the last result we have used the second part of $\left(\ref{lemm cotrad}\right)$ and $\left(\ref{PHY assumption 2}\right)$.

On the other hand, According to [7], we know that

[TABLE]

and the contradiction follows from the fact that the RHS of the estimate above goes to zero as n goes to infinity and

[TABLE]

.

**2nd case: **

The sequence $\left(t_{n}\right)$ is bounded. Setting

[TABLE]

Using the fact that the sequence $\left(t_{n}\right)$ is bounded and $\left(\ref{PHY assumption}\right),$ we infer that there exist $\alpha_{0},~{}\alpha_{1}>0,$ such that

[TABLE]

To finish the proof we need to proceed as in the first case.

The result below is a week weighted observability inequality and we need it to control the L2 norm of the solution.

Proposition 3**.**

We assume that $\omega$ and $\mathcal{O}$ satisfy the GCC and the assumption A1 holds. Let $T>\max\left(T_{\omega},T_{\mathcal{O}},T_{0}\right)$ and $\alpha>0.$ Let $\phi$ be a positive function in $C^{2}\left(\mathbb{R}_{+}\right)$ such that

[TABLE]

In addition, if $\phi^{\prime}$ is not the null function, we assume that  there exists a positive constant $K$ such that

[TABLE]

Moreover we suppose that

[TABLE]

Then there exists $C_{T,\alpha}>0,$ such that for all $\left(u_{0},v_{0},u_{1},v_{1}\right)\in\mathcal{H},$ and all $t>0,$ the solution of the system

[TABLE]

satisfies the inequality

[TABLE]

Proof.

To prove the estimate $\left(\ref{compactness lemma estimate}\right),$ we argue by contradiction. We assume that there exist a positive sequence $\left(t_{n}\right)$ and a sequence $\left(U_{n}\left(t\right)=\left(u_{n}\left(t\right),v_{n}\left(t\right),\partial_{t}u_{n}\left(t\right),\partial_{t}v_{n}\left(t\right)\right)\right)$ of solutions of the system $\left(\ref{compactness lemma system}\right)$ with initial data $\left(u_{0,n},v_{0,n},u_{1,n},v_{1,n}\right)\in\mathcal{H},$ such that

[TABLE]

**1st case: **

The sequence $\left(t_{n}\right)$ is bounded. Setting

[TABLE]

Using the fact that the sequence $\left(t_{n}\right)$ is bounded, $\left(\ref{rho assumption}\right)$ and $\left(\ref{compactness lemma contradiction estimate}\right),$ we infer that there exist $\alpha_{0},~{}\alpha_{1},~{}\alpha_{2}>0,$ such that

[TABLE]

To finish the proof we use the unique continuation hypotheses (A1) and we proceed as in [7, Proof of lemma 7].

**2nd case: **

$t_{n}\underset{n\rightarrow+\infty}{\longrightarrow}\infty.$ Setting

[TABLE]

Therefore

[TABLE]

where

[TABLE]

Thanks to $\left(\ref{compactness lemma contradiction estimate}\right)$ we get

[TABLE]

Now using the estimates above, we deduce that there exist $\alpha_{1}>0,$ such that

[TABLE]

Utilizing $\left(\ref{lemm cotrad 1}\right),\left(\ref{rho assumption 1}\right)$ and $\left(\ref{rho assumption 2}\right),$ we can show that

[TABLE]

On the other hand,  it is easy to see that

[TABLE]

noting that in the last result we have used the second part of $\left(\ref{lemm cotrad 1}\right)$ and $\left(\ref{rho assumption 2}\right)$. To finish the proof we use the unique continuation hypotheses (A1) and we proceed as in [7, Proof of lemma 7].

Lemma 2**.**

Let $\varphi\in C^{2}\left(\mathbb{R}_{+}\right)$ and non-decreasing. Let $u$ be a solution of the system $\left(\ref{system intro}\right)$ with initial data $U_{0}=\left(u_{0},v_{0},u_{1},v_{1}\right)\in D\left(\mathcal{A}\right).$ We set

[TABLE]

where $k,~{}T$ and $k_{1}$ are positive constants. Then

[TABLE]

Proof.

We differentiate $X\left(t\right)$ with respect to $t$, we obtain

[TABLE]

Using the first and the second equations of $\left(\ref{system intro}\right),$ we infer that

[TABLE]

Thanks to Young’s inequality we get

[TABLE]

To estimate the third term of the RHS of $\left(\ref{Xt derivative}\right),$ we use Poincare’s inequality and the fact that the energy is decreasing

[TABLE]

For the last term of the RHS of $\left(\ref{Xt derivative}\right),$ we use the fact that

[TABLE]

Combining the estimates above, making some arrangement and integrating the result between $t$ and $t+T$ , we obtain $\left(\ref{Xt estimate}\right).$

Let

[TABLE]

Noting that according to [4], if $h^{-1}$ is a strictly convex $C^{1}$ function from $\left[0,r_{0}\right]$ to $\mathbb{R}$ such that $h^{-1}\left(0\right)=\left(h^{-1}\right)^{\prime}\left(0\right)=0$. Then the convex conjugate function of $H$ is defined by

[TABLE]

Lemma 3**.**

We assume that the assumption A2 holds. Let $\varphi$ be a solution of the following ODE

[TABLE]

where $\epsilon_{0}$ and $C_{1}$ are positive constant. Then we have $\varphi$ is a concave strictly increasing function in $C^{3}\left(\mathbb{R}_{+}\right)$. In addition we have

1. (1)

$\underset{t\rightarrow\infty}{\lim}\varphi\left(t\right)=\infty.$** 2. (2)

$\underset{t\rightarrow\infty}{\lim}\varphi^{\prime}\left(t\right)=\frac{\epsilon_{0}\alpha_{0}}{2C_{1}}$* and $\underset{t\rightarrow\infty}{\lim}\left|\dfrac{\varphi^{\prime\prime}\left(t\right)}{\varphi^{\prime}\left(t\right)}\right|=0$.* 3. (3)

The function  $t\longmapsto\left|\frac{\varphi^{\prime\prime}\left(t\right)}{\varphi^{\prime}\left(t\right)}\right|$ is decreasing. 4. (4)

If $\varphi^{\prime\prime}$ is not the null function, then there exists $K>0$ such that $\underset{\mathbb{R}_{+}}{\sup}\left|\frac{\varphi^{\prime\prime\prime}\left(t\right)}{\varphi^{\prime\prime}\left(t\right)}\right|\leq K.$ 5. (5)

$\mathop{\displaystyle\int}_{0}^{\infty}\left|\varphi^{\prime\prime}\left(s\right)\right|ds=\varphi^{\prime}\left(0\right)-\frac{\epsilon_{0}\alpha_{0}}{2C_{1}}.$** 6. (6)

$\frac{\epsilon_{0}}{2C_{1}}\mathop{\displaystyle\int}_{0}^{\infty}\varphi\left(s\right)H^{\ast}\left(\frac{2C_{1}\varphi^{\prime}\left(s\right)}{\epsilon_{0}\varphi\left(s\right)}\right)ds\leq\frac{1}{\beta-1}\varphi^{1-\beta}\left(0\right).$**

Proof.

Using the second part of $\left(\ref{assumption2}\right)$ and the fact that

[TABLE]

we obtain

[TABLE]

for all $t\in\mathbb{R}_{+},$ which means that the function $\varphi$ is a concave on $\mathbb{R}_{+}.$

It is easy to see that the function $\varphi\in C^{3}\left(\mathbb{R}_{+}\right)$.

1. (1)

First we note that

[TABLE]

Therefore, using the fact that

[TABLE]

we deduce that

[TABLE]

thus

[TABLE] 2. (2)

Thanks to $\left(\ref{limit phi}\right)$ and $\left(\ref{assumption2}\right)$ we get

[TABLE]

Direct computations and using $\left(\ref{assumption1}\right),$ yield

[TABLE] 3. (3)

From $\left(\ref{concave phi}\right),$ we deduce that

[TABLE]

We differentiate the estimate above and making some arrangements, we obtain

[TABLE]

From the estimate above, $\left(\ref{phi t beta bound}\right)$ and $\left(\ref{assumption2}\right),$ we see that the function  $t\longmapsto\left|\frac{\varphi^{\prime\prime}\left(t\right)}{\varphi^{\prime}\left(t\right)}\right|$ is decreasing. 4. (4)

We differentiate the identity $\left(\ref{concave phi}\right)$ and making some arrangements, we obtain

[TABLE]

On the other hand, from $\left(\ref{phi equation}\right)$ and $\left(\ref{concave phi}\right),$ we infer that

[TABLE]

Combining the two estimates above, we see that

[TABLE]

So from Assumption A2, we conclude that there exists $K>0$ such that

[TABLE] 5. (5)

Using $\left(\ref{concave phi}\right)$ and $\left(\ref{limit phi derivative}\right),$ we obtain

[TABLE] 6. (6)

Thanks to $\left(\ref{convex conjugate}\right)$ and $\left(\ref{phi equation}\right),$ we see that

[TABLE]

therefore integrating the estimate above between zero and infinity and using $\left(\ref{limit phi}\right)$ and the fact that $\beta>1$ we obtain

[TABLE]

2.1. Proof of Theorem 1

We assume that $\omega$ and $\mathcal{O}$ satisfy the GCC and the assumption A1 holds. Let $\left(u,v\right)$ be a solution of the system $\left(\ref{system intro}\right)$ with initial data $U_{0}=\left(u_{0},v_{0},u_{1},v_{1}\right)\in D\left(\mathcal{A}\right).$ Let $T>\max\left(T_{\omega},T_{\mathcal{O}},T_{0}\right).$

We have $u$ is a solution of the nonhomogeneous wave equation with a localized nonlinear damping and $\left(\omega,T\right)$ satisfies the GCC. In addition, taking into account of lemma $\left(\ref{proposition phi}\right),$ we see that $\varphi^{\prime}$ satisfies the required assumptions of proposition $\left(\ref{observability lemma}\right).$ Therefore, using the observability estimate $\left(\ref{observability by a}\right)$ and $\left(\ref{Xt estimate}\right),$ we deduce that

[TABLE]

To estimate $\mathop{\displaystyle\int}_{t}^{t+T}\varphi^{\prime}\left(s\right)\mathop{\displaystyle\int}_{\Omega}\left|\partial_{t}v\left(s,x\right)\right|^{2}dxds,$ we first use the fact that $v$ is a solution of the nonhomogeneous wave equation and $\left(\mathcal{O},T\right)$ satisfies the GCC, then from the observability estimate $\left(\ref{observability by a}\right),$ we infer that

[TABLE]

Now we estimate $\mathop{\displaystyle\int}_{t}^{t+T}\varphi^{\prime}\left(s\right)\mathop{\displaystyle\int}_{\Omega}b\left(x\right)\left|\partial_{t}v\left(s,x\right)\right|^{2}dxds.$ We have

[TABLE]

We multiply the first equation of $\left(\ref{system derivative}\right)$ by $\left(\varphi^{\prime}\left(t\right)\partial_{t}v\right)$ and the second equation by $\left(\varphi^{\prime}\left(t\right)\partial_{t}u\right)$ and integrating the difference of these results over $\Omega$, we obtain

[TABLE]

Using Young’s inequality, we infer that

[TABLE]

Integrating the estimate above between $t$ and $t+T,$ we obtain

[TABLE]

Now using the observability estimate $\left(\ref{proof theorem estimate dtv}\right)$ and taking $\epsilon=\frac{1}{4\left\|a\right\|_{\infty}C_{T}},$ we get

[TABLE]

Combining the estimate above and $\left(\ref{proof theorem estimate dtv}\right),$ we find that

[TABLE]

Now using the observability estimate $\left(\ref{observability by a}\right),$ we infer that

[TABLE]

Utilizing $\left(\ref{proof theorem Xt estimate}\right),$ $\left(\ref{proof theorem estimate dtv final}\right)$ and making some arrangements, we obtain

[TABLE]

We take $k_{1}=8C_{T},$ we conclude that

[TABLE]

Now using $\left(\ref{compactness lemma estimate}\right)$ with $\alpha=\frac{1}{2\left(\left\|a\right\|_{\infty}+12C_{T}\left\|b\right\|_{\infty}+32C_{T}^{2}\left\|b\right\|_{\infty}^{2}\right)}$ , we get

[TABLE]

where

[TABLE]

Now we have to estimate the first term of RHS of the estimate above by the third term of the LHS. We set for all fixed $s\geq 0,~{}\Omega^{s}=\left\{x,\left|\partial_{t}u\left(s,x\right)\right|<1\right\}.$ Thanks to $\left(\ref{h0 defin}\right),$ we have

[TABLE]

Since $h_{0}$ is concave, we can use (the reverse) Jensen’s inequality and we obtain

[TABLE]

Under our assumptions the function $H$  defined by $\left(\ref{H definition}\right)$ is convex and proper. Hence, we can apply Young’s inequality [19]

[TABLE]

where $H^{\ast}$ is the convex conjugate of the function $H.$

On the other hand, using the fact that the function $g$ is linearly bounded near infinity, we infer that

[TABLE]

The estimate above combined with $\left(\ref{proof theorem Xt final estimate1}\right)$, give

[TABLE]

We remind that

[TABLE]

with $k_{1}=8C_{T}.$ Using Young’s inequality and Poincaré inequality, it is easy to see that

[TABLE]

So, taking $k$ such that

[TABLE]

using $\left(\ref{proof theorem Xt final estimate}\right)$ and the fact that

[TABLE]

we deduce that $X\left(t\right)\geq 0$ and

[TABLE]

for all $t\geq 0$. Thus

[TABLE]

and this gives

[TABLE]

Utilizing lemma 3, we conclude that there exists a positive constant $C$ such that

[TABLE]

Since

[TABLE]

then $\left(\ref{integral bound}\right),$ gives $\left(\ref{energy decay rate}\right).$

Finally, using the density of $D\left(\mathcal{A}\right)$ in $\mathcal{H}$, we obtain $\left(\ref{limit energy space}\right).$