Exponential decay for the semilinear wave equation with localized frictional and Kelvin

Abstract

In the present paper, we are concerned with the semilinear viscoelastic wave equation in an inhomogeneous medium Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space.

Keywords

Wave equation Kelvin–Voigt damping frictional damping source term

1. Introduction

1.1. Description of the problem

This article is devoted to the analysis of the exponential and uniform decay rates of solutions to the wave equation subject to two localized dampings: $\begin{matrix} (1.1) & \{\begin{matrix} ρ (x) \partial_{t}^{2} u - div (K (x) \nabla u) + f (u) - div (a (x) \nabla \partial_{t} u) + b (x) \partial_{t} u = 0 & in Ω \times (0, \infty), \\ u = 0 & on \partial Ω \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x) & in Ω, \end{matrix} \end{matrix}$ where Ω is a bounded domain of $R^{n}$ for $n ⩾ 2$ , with smooth boundary $\partial Ω$ , $ρ : Ω \to R_{+}$ , $k_{i j} : Ω \to R$ , $1 ⩽ i$ , $j ⩽ n$ are $C^{\infty} (Ω)$ functions such that for all $x \in Ω$ and $ξ \in R^{n}$ , $\begin{matrix} (1.2) & α_{0} ⩽ ρ (x) ⩽ β_{0}, k_{i j} (x) = k_{j i} (x), α | ξ |^{2} ⩽ ξ^{⊤} \cdot K (x) \cdot ξ ⩽ β | ξ |^{2}, \end{matrix}$ where $α_{0}$ , $β_{0}$ , α, β are positive constants and $K (x) = {(k_{i j})}_{i, j}$ is a symmetric positive-definite matrix. We denote by ω, with smooth boundary $\partial ω$ , the intersection of Ω with a neighborhood of $\partial Ω$ in $R^{n}$ and call it a neighborhood of the boundary of Ω. In addition, $a = a (x) \in L^{\infty} (Ω) \cap C^{0} (\overline{ω})$ , is a real valued function.

The function f satisfies the following hypotheses:

Assumption 1.1.
$f : R \to R$ is a $C^{2}$ function with sub-critical growth, that is: $\begin{matrix} (1.3) & f (0) = 0, | f^{(j)} (s) | ⩽ k_{0} {(1 + | s |)}^{p - j}, for all s \in R and j = 1, 2 . \end{matrix}$ In particular, we obtain from (1.3), $\begin{matrix} (1.4) & | f (r) - f (s) | ⩽ c (1 + | s |^{p - 1} + | r |^{p - 1}) | r - s |, for all s, r \in R, \end{matrix}$ for some $c > 0$ , with $\begin{matrix} (1.5) & 1 ⩽ p < \frac{n}{n - 2} if n ⩾ 3 or p ⩾ 1 if n = 1, 2 . \end{matrix}$ Moreover, $\begin{matrix} (1.6) & 0 ⩽ F (s) ⩽ f (s) s, for all s \in R, \end{matrix}$ where $F (λ) : = \int_{0}^{λ} f (s) d s$
Assumption 1.2.
ω geometrically controls Ω, i.e. there exists $T_{0} > 0$ , such that every geodesic of the metric $G (x)$ , where $G (x) = {(\frac{K (x)}{ρ (x)})}^{- 1}$ travelling with speed 1 and issued at $t = 0$ , intercepts ω in a time $t < T_{0}$ .

Assumption 1.2 is the so called Geometric Control Condition (G.C.C.). It is well-known that it is necessary and sufficient for stabilization and control of the linear wave equation, see [2,5,7,8,13,29] and the references therein. For this reason and since in the present paper we do not have any control of the geodesics because of the inhomogeneous medium we consider ω a neighbourhood of the whole boundary $\partial Ω$ and give conditions on the metric $G = {(K / ρ)}^{- 1}$ in order every geodesic of the metric G enters the set ω at a time $t < T_{0}$ .
Remark 1.1.
It is important to observe that Assumption 1.2 is not obviously fulfilled for every matrix $G = {(K / ρ)}^{- 1}$ . For instance, the equator in the unit sphere $S^{2}$ is a periodic geodesic and we can consider $Ω \subset S^{2}$ that contains the equator as a domain in $R^{2}$ endowed with a Riemannian metric G. In [9], the authors give examples where this situation occurs, that is, it is possible to choose the density function $ρ (x)$ so that every geodesic curve (in black) of the metric $G = {(K / ρ)}^{- 1}$ enters in the damped area (in gray) satisfying the Geometric Control Condition according to Fig. 1. Thus, uniform decay rate estimates are expected in this case. An easy one happens when $G (x) = I_{d}$ . In this case the geodesics are straight lines and necessarily they will meet the region ω.

When we do not have any control on the geodesics of the metric $G = {(K / ρ)}^{- 1}$ , we have to assume damping everywhere on Ω, satisfying the following assumptions:
for all $x \in \partial Ω$ , $a (x) > 0$ ;

for all geodesic $t \in I \mapsto x (t) \in Ω$ of the metric $G = {(K / ρ)}^{- 1}$ , with $0 \in I$ , there exists $t ⩾ 0$ such that $a (x (t)) > 0$ .
The best way to do this is by using the ideas introduced in Cavalcanti et al. [7,8], namely, $a (x) ⩾ a_{0}$ in a neighborhood, ω, of the boundary $\partial Ω$ , while $a (x) ⩾ a_{0}^{*} > 0$ in $(Ω ∖ ω) ∖ V$ , where $V = ⋃_{i = 1}^{k} V_{i}$ and $meas (V) ⩾ meas (Ω ∖ ω) - ε$ , for an arbitrary $ε > 0$ . In Fig. 2, the demarcated region ω (in gray) and $(Ω ∖ ω) ∖ V$ (in light gray) illustrate the damped region on the manifold $(Ω, G)$ , which can be considered with measure as small as desired, however totally distributed on Ω. The demarcated region $V : = ⋃_{i = 1}^{k} V_{i}$ (in white) illustrates the region without damping with measure arbitrarily large but also totally distributed in Ω.

Fig. 1.
Geodesic entering the damped area.

Fig. 2.
Totally distributed damping.

The following assumptions are made on the functions $a (x)$ , $b (x)$ , responsible for the localized dissipative effect:
Assumption 1.3.
We assume that $a (\cdot) \in L^{\infty} (Ω)$ is a nonnegative function. In addition, there exists a compact, connected set $A \subset Ω$ with smooth boundary and non-empty interior, verifying $\begin{matrix} A : = {x \in Ω : a (x) = 0} . \end{matrix}$ We also assume that $a \in C^{0} (\overline{ω})$ , where $ω : = Ω ∖ A$ .
Assumption 1.4.
$b \in C^{0} (\overline{Ω})$ is a nonnegative function such that $b (x) ⩾ b_{0} > 0$ in a neighbourhood of the boundary $\partial A$ of the set $A : = {x \in Ω : a (x) = 0}$ , according to Fig. 3 and Fig. 4.

One of the main ingredients for the stabilization of the system (1.1) is a Unique Continuation Principle, so we assume the following hypothesis:

Fig. 3.
The Kelvin–Voigt damping $a (x)$ is positive in $ω : = Ω ∖ A$ while the frictional damping $b (x)$ is effective in a neighbourhood of $\partial A$ , that is, $b (x) ⩾ b_{0} > 0$ in $V_{ε} = {x \in Ω : d (x, y) < ε, y \in \partial A}$ for $ε > 0$ small enough if $G = {(K / ρ)}^{- 1} = I_{d}$ or according to Fig. 1.

Fig. 4.
Admissible geometries for the Kelvin–Voigt and frictional dissipations $a (x)$ and $b (x)$ , respectively, if $G = {(K / ρ)}^{- 1} = I_{d}$ or according to Fig. 1.
Assumption 1.5.
For every $T ⩾ T_{0}$ , the only solution $u \in C (] 0$ , $T [; L^{2} (Ω)) \cap C (] 0, T [, H^{- 1} (Ω))$ to the problem $\begin{matrix} (1.7) & \{\begin{matrix} ρ (x) u_{t t} - div (K (x) \nabla u) + V (x, t) u = 0 & in Ω \times (0, T), \\ u = 0 & on ω, \end{matrix} \end{matrix}$ where $V (x, t) \in L^{\infty} (] 0, T [, L^{n} (Ω))$ , is the trivial one $u = 0$ .
Remark 1.2.
This type of condition in Assumption 1.5 is generally assumed in control/stabilization statements and the possibility to overcome it to every setting seems to be an open question. However, there is concrete examples where Assumption 1.5 holds, for example, if $V \equiv 0$ it is satisfied by the results in the lecture notes of Burq and Gérard (see equations (6.28) and (6.29) on page 75 of [6]). One can also give an example where the unique continuation principle holds globally and also applies to the Laplacian with coefficients, see for instance Section 2.1.2 and Theorem 2.2 in [14]. Therefore, when $ρ = 1$ , Theorem 2.2 in [14] provides an example where the Assumption 1.5 holds. For more details see [19,30] and [35].

The paper is organized as follows. In Section 2 we give some notation and we establish the well-posedness to problem (1.1). In Section 3 we give the proof of the stabilization which consists our main result. In the Appendix we recall some basic results of microlocal analysis.
1.2. Previous results, main goal and methodology

The decay properties of solutions to the wave equation have been widely studied by many authors under different conditions. Among the numerous papers regarding the wave equation we mention the following references: [1–3,5,7,8,11–13,16–18,20–22,25–27,29,30,32,34] and [36] and a long list of references therein.

The study of problem (1.1) presents two main points of difficulty. The first one is to deal with the viscoelastic damping of Kelvin–Voigt type, which generates an unbounded operator. In addition, the domain consists of two different materials, that is, there is an interaction between the elastic component (the portion of Ω where $a \equiv 0$ ) and the other one which is the viscoelastic component (the portion of Ω where $a > 0$ ).

When $f \equiv 0$ , the damped wave equation subject to a locally distributed viscoelastic effect of Kelvin–Voigt type has been studied, in the existing literature, by many authors, for instance, [4,23,24,33] and references therein. In [23], Liu and Liu consider the problem posed in an interval $(0, L)$ , $0 < L < + \infty$ , and they prove that if the damping coefficient $a = a (x) \in L^{\infty} (0, L)$ is effective in a subset $(α, β)$ , such that $0 < α < β < L$ , the energy does not decay uniformly.

In [10], the authors studied an equation of the type $y_{t t} - Δ y + a (x) y_{t} - div (b (x) \nabla y_{t}) = 0$ and by using a combination of the multiplier techniques and the frequency domain method, they show that a convenient interaction of the two damping mechanisms is powerful enough for the exponential stability of the dynamical system, provided that the coefficient of the Kelvin–Voigt damping is smooth enough and satisfies a structural condition.

In [24], Liu and Rao study the problem posed in higher dimensions and, even considering more regular damping coefficient and smooth initial data, they conclude that there is a loss of regularity which makes it difficult to apply the multiplier method. To bypass these difficulties, the authors were forced to make several technical assumptions on the damping coefficient. Tebou in [33], relaxed the damping coefficients hypothesis as well as the conditions on the feedback control region, but he had to impose a certain constraint on the gradient of the damping coefficient, which will not be required in our current study.

The main contribution of the present paper is to prove the exponential stability of problem (1.1), which generalizes the previous results given in [10] to the semilinear case in the inhomogeneous medium Ω. In particular, the method allows us to consider an inhomogeneous medium subject to a Kelvin–Voigt type damping acting in a neighborhood of the boundary or in a mesh totally distributed in the domain with measure arbitrarily small.

In order to obtain the desired stability result for the wave equation subject to the Kelvin–Voigt and frictional dampings, we combined the presence of the two dissipations in order to use some results present in the literature. The method of proof combines an observability inequality, microlocal analysis tools and unique continuation properties. We also emphasize that the usual multiplier technique, $2 m \cdot \nabla u + (n - 1) u$ , does not apply to this equation, since there is a loss of regularity of the solutions even for regular initial data and also due to the non-linear character of the problem.

In what follows we are going to explain briefly the methodology we are going to use.

In the presence of both dissipative effects, the energy identity associated to problem (1.1) is given by $\begin{array}{l} E_{u} (t) + \int_{0}^{t} \int_{Ω} a (x) {| \nabla \partial_{t} u (x, s) |}^{2} d x d s \\ (1.8) & + \int_{0}^{t} \int_{Ω} b (x) {| \partial_{t} u (x, s) |}^{2} d x d s = E_{u} (0), for all t \in [0, + \infty), \end{array}$ where $\begin{matrix} (1.9) & E_{u} (t) : = \frac{1}{2} \int_{Ω} ρ (x) {| \partial_{t} u (x, t) |}^{2} + {(\nabla u (x, t))}^{⊤} \cdot K (x) \cdot \nabla u (x, t) d x + \int_{Ω} F (u (x, t)) d x, \end{matrix}$ with $F (s) : = \int_{0}^{s} f (λ) d λ$ . Furthermore, we will also prove the corresponding observability inequality to problem (1.1), that is, we shall prove that there exists a positive constant C, verifying $\begin{matrix} (1.10) & E_{u} (0) ⩽ C \int_{0}^{T} \int_{Ω} (b (x) | \partial_{t} u |^{2} + a (x) | \nabla \partial_{t} u |^{2}) d x d t, for all T ⩾ T_{0} . \end{matrix}$

In what follows, given $δ > 0$ , $V_{δ}$ will denote the neighbourhood of $\partial A$ given by $\begin{matrix} V_{δ} : = {x \in Ω : d (x, y) < δ, y \in \partial A} . \end{matrix}$

It is worth mentioning that if $a (x) = 0$ , for all $x \in Ω$ , the frictional damping $b (x)$ is not strong enough to provide the exponential and uniform decay of the energy since in this case the geometric control condition (GCC) is violated, see references [2,5]. However, the frictional damping plays a key role which we will describe next.

Let $C : = {x \in A : d (x, y) > ε / 2, y \in \partial A}$ . To prove (1.10) and therefore the stability result, we will argue by contradiction and we will obtain a sequence ${u^{m}}_{m \in N}$ of solutions to problem (1.1) such that $E_{u^{m}} (0) = 1$ . The contradiction is obtained proving that $E_{u^{m}} (0) \to 0$ as $m \to + \infty$ . By exploiting the properties of $a (\cdot)$ and $b (\cdot)$ and the Poincaré inequality, we shall prove that $\begin{matrix} (1.11) & \int_{0}^{T} \int_{Ω ∖ C} {| \partial_{t} u^{m} |}^{2} d x d t \to 0, as m \to + \infty . \end{matrix}$ To propagate the convergence (1.11) to the whole set $Ω \times (0, T)$ , we use microlocal analysis arguments. Indeed, we consider the microlocal defect measure μ, associated to ${\partial_{t} u^{m}}_{m \in N}$ . First, we shall establish the convergence $\begin{matrix} ρ (x) \partial_{t}^{2} \partial_{t} u^{m} - div (K (x) \nabla (\partial_{t} u^{m})) \to 0 strongly in H_{loc}^{- 2} (Ω \times (0, T)), as m \to + \infty, \end{matrix}$ which is enough to ensure that the $supp (μ)$ is contained in the characteristic set of the wave operator However, the last convergence is not sufficient for propagation, since we need a stronger convergence, namely, $\begin{matrix} ρ (x) \partial_{t}^{2} \partial_{t} u^{m} - div (K (x) \nabla (\partial_{t} u^{m})) \to 0 strongly in H_{loc}^{- 1} (Ω \times (0, T)), as m \to + \infty . \end{matrix}$ The problematic term, responsible for this lack of regularity, is $div (a (x) \nabla \partial_{t} u^{m})$ , because $\begin{matrix} \partial_{t} (div (a (x) \nabla \partial_{t} u^{m})) \to 0 strongly in H_{loc}^{- 2} (Ω \times (0, T)), as m \to + \infty . \end{matrix}$

To circumvent this difficulty, we use the fact that the frictional damping is effective in the neighbourhood $V_{ε}$ of $\partial A$ . Note that $a (x) \nabla \partial_{t} u^{m} = 0$ in $A \times (0, T)$ , consequently, since $\begin{matrix} ρ (x) \partial_{t}^{2} u^{m} - div (K (x) \nabla (u^{m})) = - f (u^{m}) - b (x) \partial_{t} u^{m} in A \times (0, T) . \end{matrix}$ Assuming for a moment that $f (u^{m}) \to 0$ in $L^{2} (Ω \times (0, T))$ as $m \to + \infty$ , we obtain $\begin{matrix} (1.12) & □ \partial_{t} u^{m} \to 0 in H_{loc}^{- 1} (int (A) \times (0, T)), as m \to + \infty . \end{matrix}$

From convergence (1.12) we deduce that μ propagates along the bicharacteristic flow of the operator $□ (\cdot)$ , which means that if there is $ω_{0} = (t_{0}, x_{0}, τ_{0}, ξ_{0})$ such that $ω_{0} = (t_{0}, x_{0}, τ_{0}, ξ_{0}) \notin supp (μ)$ , the whole bicharacteristic issued from $ω_{0}$ does not belong to $supp (μ)$ . However, since $supp (μ) \subset C \times (0, T) \subset A \times (0, T)$ , and the frictional damping acts in both sides of the boundary $\partial A$ , we can propagate the kinetic energy from $(V_{ε / 2} \cap A) \times (0, T)$ towards the set $C \times (0, T)$ .

Consequently, we get that $supp (μ) = \emptyset$ and therefore, $\partial_{t} u^{m} \to 0$ in $L_{loc}^{2} (Ω \times (0, T))$ . With some calculations we can show that $\begin{matrix} (1.13) & \int_{0}^{T} \int_{Ω} {| \partial_{t} u^{m} (x, t) |}^{2} d x d t \to 0, as m \to + \infty . \end{matrix}$

In light of convergence (1.13) and an argument of equipartition of energy, we can conclude that $E_{u^{m}} (0) \to 0$ , as $m \to + \infty$ , obtaining the desired contradiction. This will be clarified in Section 3.

2. Well-posedness

We consider the weak phase space $\begin{matrix} H = H_{0}^{1} (Ω) \times L^{2} (Ω), \end{matrix}$ which is endowed with the inner product $\begin{matrix} {⟨ (u_{1}, v_{1}), (u_{2}, v_{2}) ⟩}_{H} = \int_{Ω} (\nabla u_{1}^{T} \cdot K (x) \cdot \nabla u_{2} + ρ v_{1} v_{2}) d x . \end{matrix}$ Denoting $v = u_{t}$ we may rewrite problem (1.1) as the following Cauchy problem in $H$ $\begin{matrix} (2.1) & \{\begin{matrix} \frac{\partial}{\partial t} (u, v) = A (u, v) + F (u, v), \\ (u, v) (0) = (u_{0}, v_{0}), \end{matrix} \end{matrix}$ where the linear unbounded operator $A : D (A) \to H$ is given by $\begin{matrix} (2.2) & A (u, v) = (v, \frac{1}{ρ} div (K (x) \nabla u + a (x) \nabla v) - \frac{1}{ρ} b (x) v), \end{matrix}$ with domain $\begin{matrix} (2.3) & D (A) = {(u, v) \in H : v \in H_{0}^{1} (Ω), div (K (x) \nabla u + a (x) \nabla v) \in L^{2} (Ω)} \end{matrix}$ and $F : H \to H$ is the nonlinear operator $\begin{matrix} (2.4) & F (u, v) = (0, - \frac{1}{ρ} f (u)) . \end{matrix}$ Now, we are in conditions to state the well-posedness result for problem (2.1), which ensures that problem (1.1) is globally well-posed.

Theorem 2.1 (Global Well-Posedness).

Assume that the hypotheses on the nonlinear term f specified in Assumption 1.1 are satisfied and the initial data $(u_{0}, v_{0}) \in H$ . Then problem ( 2.1 ) possesses a unique mild solution $(u, v) \in C ([0, T]; H)$ .

Moreover, if $(u_{0}, v_{0}) \in D (A)$ , then the solution is regular.

Proof.
First of all the operator $A : D (A) \subset H \to H$ defined by (2.2) and (2.3) generates a $C_{0}$ -semigroup of contractions $e^{A t}$ on the energy space $H$ by the results of Liu and Rao in [24]. Moreover the nonlinear operator $F : H \to H$ given in (2.4) is a locally Lipschitz continuous operator. Indeed, given a bounded set B in $H$ and $(u_{1}, v_{1}), (u_{2}, v_{2}) \in B$ . Then, by Assumption 1.1, using Hölder’s inequality to the conjugate exponents p, $p / (p - 1)$ and the fact that $H^{1}$ is continuously embedded in $L^{2 p} (Ω)$ , we obtain $\begin{array}{rcl} {‖ F (u_{1}, v_{1}) - F (u_{2}, v_{2}) ‖}_{H}^{2} & ⩽ & {‖ f (u_{1}) - f (u_{2}) ‖}_{L^{2}}^{2} \\ ⩽ & C (1 + ‖ u_{1} ‖_{L^{2 p}}^{2 (p - 1)} + ‖ u_{2} ‖_{L^{2 p}}^{2 (p - 1)}) ‖ u_{1} - u_{2} ‖_{L^{2 p}}^{2} \\ ⩽ & C_{0} ‖ u_{1} - u_{2} ‖_{H^{1}}^{2} ⩽ C_{0} {‖ (u_{1}, v_{1}) - (u_{2}, v_{2}) ‖}_{H}^{2} . \end{array}$ Hence by Theorems 6.1.4 and 6.1.5 in Pazy’s book [28], the Cauchy problem (2.1) has a unique mild solution $\begin{matrix} (u (t), v (t)) = e^{A t} (u_{0}, v_{0}) + \int_{0}^{t} e^{A (t - s)} F (u (s), v (s)) d s, t \in [0, T_{\max}) . \end{matrix}$ Let us see that $T_{\max} = \infty$ . Indeed, the energy functional associated to problem (1.1) is given by $\begin{matrix} (2.5) & E_{u} (t) : = \frac{1}{2} \int_{Ω} ρ (x) {| \partial_{t} u (x, t) |}^{2} + {(\nabla u)}^{⊤} \cdot K (x) \cdot \nabla u d x d t + \int_{Ω} F (u (x, t)) d x, \end{matrix}$ where $F (λ) = \int_{0}^{λ} f (s) d s$ . Thus, it follows that $\begin{matrix} (2.6) & E_{u} (t_{1}) - E_{u} (t_{2}) = - \int_{t_{1}}^{t_{2}} \int_{Ω} a (x) {| \nabla u_{t} (x, t) |}^{2} + b (x) | \partial_{t} u |^{2} d x . \end{matrix}$ Indeed, the identity (2.6) is true for regular solutions, and the final desired equality follows by a density argument. Thus, from (2.6) we deduce that $E_{u} (t)$ is non-increasing with $E_{u} (t) ⩽ E_{u} (0)$ for all $t \in [0, T_{\max})$ . On the other hand, from (1.6), we deduce $\begin{matrix} E_{u} (t) ⩾ \frac{1}{2} \int_{Ω} ρ (x) {| \partial_{t} u (x, t) |}^{2} d x + \frac{1}{2} \int_{Ω} {(\nabla u (x, t))}^{⊤} \cdot K (x) \cdot \nabla u (x, t) d x ⩾ \frac{1}{2} {‖ (u, u_{t}) ‖}_{H}^{2}, \end{matrix}$ that is $\begin{matrix} \frac{1}{2} {‖ (u, u_{t}) ‖}_{H}^{2} ⩽ E_{u} (t) ⩽ E_{u} (0), \forall t \in [0, T_{\max}) . \end{matrix}$ Therefore the local solutions cannot blow-up in finite time and it follows that $T_{\max} = \infty$ . □

3. Exponential decay to problem (1.1)

Let $T_{0} > 0$ be associated to the geometric control condition, that is, every ray of the geometric optics enters $ω : = Ω ∖ A$ in a time $T^{*} < T_{0}$ .

The energy identity reads as follows $\begin{matrix} (3.1) & E_{u} (t_{2}) - E_{u} (t_{1}) = - \int_{t_{1}}^{t_{2}} \int_{Ω} a (x) | \nabla \partial_{t} u |^{2} + b (x) | \partial_{t} u |^{2} d x d t, \end{matrix}$ for all $0 ⩽ t_{1} ⩽ t_{2} < + \infty$ .

Thus, our goal is to prove the observability inequality established in the following lemma.

Lemma 3.1.

For all $T ⩾ T_{0}$ and all $R > 0$ , there exists a positive constant $C > 0$ such that the corresponding solution u to ( 1.1 ) satisfies the inequality $\begin{matrix} (3.2) & E_{u} (0) ⩽ C (\int_{0}^{T} \int_{Ω} a (x) | \nabla \partial_{t} u |^{2} d x d t + \int_{0}^{T} \int_{Ω} b (x) | \partial_{t} u |^{2} d x d t), \end{matrix}$ if the initial data satisfies $E_{u} (0) < R$ .

Proof.

Our proof relies on contradiction arguments. So, if (3.2) is false, then there exists $T ⩾ T_{0}$ and $R > 0$ such that for every constant $C > 0$ , there exists a solution $u^{C}$ to (1.1) verifying $E_{u^{C}} (0) ⩽ R$ and such that $u^{C}$ violates (3.2).

In particular, for each $m \in N$ , we obtain the existence of a solution $u^{m}$ to problem (1.1) verifying $E_{u^{m}} (0) ⩽ R$ , whose corresponding solution satisfies $\begin{matrix} (3.3) & E_{u^{m}} (0) > m (\int_{0}^{T} \int_{Ω} a (x) {| \nabla \partial_{t} u^{m} |}^{2} d x d t + \int_{0}^{T} \int_{Ω} b (x) {| \partial_{t} u^{m} |}^{2} d x d t) . \end{matrix}$

Then, we obtain a sequence ${u^{m}}_{m \in N}$ of solutions to problem (1.1) such that $\begin{matrix} lim_{m \to + \infty} \frac{E_{u^{m}} (0)}{\int_{0}^{T} \int_{Ω} (a (x) | \nabla \partial_{t} u^{m} |^{2} + b (x) | \partial_{t} u^{m} |^{2}) d x d t} = + \infty . \end{matrix}$

Equivalently $\begin{matrix} (3.4) & lim_{m \to + \infty} \frac{\int_{0}^{T} \int_{Ω} (a (x) | \nabla \partial_{t} u^{m} |^{2} + b (x) | \partial_{t} u^{m} |^{2}) d x d t}{E_{u^{m}} (0)} = 0 . \end{matrix}$

Since ${E_{u^{m}} (0)}_{m \in N}$ is bounded, (3.4) yields $\begin{matrix} (3.5) & lim_{m \to + \infty} \int_{0}^{T} \int_{Ω} (a (x) {| \nabla \partial_{t} u^{m} |}^{2} + b (x) {| \partial_{t} u^{m} |}^{2}) d x d t = 0, \end{matrix}$ which implies, by a variant of Poincaré’s inequality, that $\begin{matrix} (3.6) & lim_{m \to \infty} \int_{0}^{T} \int_{ω} {| \partial_{t} u^{m} (x, t) |}^{2} d x d t = 0 . \end{matrix}$ Furthermore, there exists a subsequence of ${u^{m}}_{m \in N}$ , still denoted by ${u^{m}}$ , verifying the following convergences: $\begin{array}{l} (3.7) & u^{m} ⇀ u weakly-star in L^{\infty} (0, T; H_{0}^{1} (Ω)), as m \to + \infty, \\ (3.8) & \partial_{t} u^{m} ⇀ \partial_{t} u weakly-star in L^{\infty} (0, T; L^{2} (Ω)), as m \to + \infty, \\ (3.9) & u^{m} \to u strongly in L^{\infty} (0, T; L^{q} (Ω)), as m \to + \infty, for all q \in [2, \frac{2 n}{n - 2}), \end{array}$ where the last convergence is obtained using Aubin–Lions–Simon Theorem (see [31]).

Consequently $\begin{matrix} (3.10) & f (u^{m}) ⇀ f (u) weakly in L^{2} (0, T; L^{2} (Ω)) . \end{matrix}$

Let $C : = {x \in A : d (x, y) > ε / 2, y \in \partial A}$ . The proof is divided into two distinguished cases: $u \neq 0$ and $u = 0$ .

Case (a): $u \neq 0$ .

For $m \in N$ , ${u^{m}}$ is the solution to the problem $\begin{matrix} \{\begin{matrix} ρ (x) \partial_{t}^{2} u^{m} - div (K (x) \nabla (u^{m}) + f (u^{m}) - div (a (x) \nabla \partial_{t} u^{m}) + b (x) \partial_{t} u^{m} = 0 & in Ω \times (0, + \infty), \\ u^{m} = 0 & on \partial Ω \times (0, + \infty), \\ u^{m} (x, 0) = u_{0, m} (x); \partial_{t} u^{m} (x, 0) = u_{1, m} (x), & in Ω . \end{matrix} \end{matrix}$

Taking (3.5), (3.7), (3.8), (3.9) and (3.10) into consideration we obtain $\begin{matrix} (3.11) & \{\begin{matrix} ρ (x) \partial_{t}^{2} u - div (K (x) \nabla (u)) + f (u) = 0 & in Ω \times (0, + \infty), \\ u = 0 & on \partial Ω \times (0, + \infty), \\ \partial_{t} u = 0 & a.e. in Ω ∖ C . \end{matrix} \end{matrix}$

Defining $y = \partial_{t} u$ , the above problem yields $\begin{matrix} \{\begin{matrix} ρ (x) \partial_{t}^{2} y - div (K (x) \nabla (y)) + f^{'} (u) y = 0 & in Ω \times (0, + \infty), \\ y = 0 & on \partial Ω \times (0, + \infty), \\ y = 0 & a.e. in Ω ∖ C . \end{matrix} \end{matrix}$

From Assumption 1.5 we deduce that $y = \partial_{t} u \equiv 0$ . Returning to (3.11) we conclude that $u \equiv 0$ as well and we obtain the desired contradiction.

Case (b): $u = 0$ .

Setting $\begin{array}{rcl} (3.12) & α_{m} : = \sqrt{E_{u^{m}} (0)}, and v^{m} : = \frac{u^{m}}{α_{m}}, \end{array}$ in light of (3.4), we obtain $\begin{matrix} (3.13) & lim_{m \to + \infty} \int_{0}^{T} \int_{Ω} (a (x) {| \nabla \partial_{t} v^{m} |}^{2} + b (x) {| \partial_{t} v^{m} |}^{2}) d x d t = 0 . \end{matrix}$

According to (3.12), for each $m \in N$ , $v^{m}$ is the solution to the following problem: $\begin{matrix} (3.14) & \{\begin{matrix} ρ (x) \partial_{t}^{2} v^{m} - div (K (x) \nabla (v^{m})) + \frac{1}{α_{m}} f (u^{m}) \\ - div (a (x) \nabla \partial_{t} v^{m}) + b (x) \partial_{t} v^{m} = 0 & in Ω \times (0, + \infty), \\ v^{m} = 0 & on \partial Ω \times (0, + \infty), \\ v^{m} (x, 0) = \frac{u_{0, m}}{α_{m}}; \partial_{t} v^{m} (x, 0) = \frac{u_{1, m}}{α_{m}} & in Ω, \end{matrix} \end{matrix}$ and the associated energy functional is given by $\begin{matrix} E_{v^{m}} (t) = \frac{1}{2} \int_{Ω} (ρ (x) {| \partial_{t} v^{m} |}^{2} + {(\nabla v^{m})}^{⊤} \cdot K (x) \cdot \nabla v^{m}) d x + \frac{1}{α_{m}^{2}} \int_{Ω} F (u^{m}) d x, \end{matrix}$ since $\begin{matrix} \frac{1}{α_{m}} \int_{Ω} f (u^{m}) \partial_{t} v^{m} d x = \frac{1}{α_{m}^{2}} \frac{d}{d t} \int_{Ω} F (u^{m}) d x . \end{matrix}$

Note that $E_{v^{m}} (t) = \frac{1}{α_{m}^{2}} E_{u^{m}} (t)$ for all $t ⩾ 0$ and, in particular, for $t = 0$ $\begin{matrix} (3.15) & E_{v^{m}} (0) = \frac{1}{α_{m}^{2}} E_{u^{m}} (0) = 1, for all m \in N . \end{matrix}$

In order to achieve the contradiction we are going to prove that $\begin{matrix} (3.16) & lim_{m \to + \infty} E_{v^{m}} (0) = 0 . \end{matrix}$

Indeed, initially, we observe that (3.15) yields the existence of a subsequence of ${v^{m}}_{m \in N}$ , reindexed again by ${v^{m}}$ , such that $\begin{array}{l} (3.17) & v^{m} ⇀ v weakly-star in L^{\infty} (0, T; H_{0}^{1} (Ω)), as m \to + \infty, \\ (3.18) & \partial_{t} v^{m} ⇀ \partial_{t} v weakly-star in L^{\infty} (0, T; L^{2} (Ω)), as m \to + \infty, \\ (3.19) & v^{m} \to v strongly in L^{\infty} (0, T; L^{q} (Ω)), as m \to + \infty, for all q \in [2, \frac{2 n}{n - 2}) . \end{array}$

For some eventual subsequence, we have that $α_{m} \to α$ with $α ⩾ 0$ .

If $α > 0$ , by (3.10) we know that $\begin{matrix} f (u^{m}) ⇀ f (u) weakly in L^{2} (0, T; L^{2} (Ω)) . \end{matrix}$ Since we are assuming that $u = 0$ and $f (0) = 0$ this translates to $\begin{matrix} f (u^{m}) ⇀ 0 weakly in L^{2} (0, T; L^{2} (Ω)) . \end{matrix}$ Furthermore, since $v^{m} : = \frac{u^{m}}{α_{m}}$ and we are assuming that $α > 0$ we obtain $\begin{matrix} (3.20) & \frac{1}{α_{m}} f (α_{m} v^{m}) ⇀ 0 weakly in L^{2} (0, T; L^{2} (Ω)) . \end{matrix}$ Thus, passing to the limit in (3.14) and considering convergences (3.13), (3.17)–(3.19), (3.20), we deduce $\begin{matrix} (3.21) & \{\begin{matrix} ρ (x) \partial_{t}^{2} v - div (K (x) \nabla (v)) = 0 & in Ω \times (0, + \infty), \\ v = 0 & on \partial Ω \times (0, + \infty), \\ \partial_{t} v = 0 & in Ω ∖ C . \end{matrix} \end{matrix}$

The above problem yields, for $w = \partial_{t} v$ , in the distributional sense, $\begin{matrix} (3.22) & \{\begin{matrix} ρ (x) \partial_{t}^{2} w - div (K (x) \nabla (w)) = 0 & in Ω \times (0, + \infty), \\ w = 0 & on \partial Ω \times (0, + \infty), \\ w = 0 & in Ω ∖ C . \end{matrix} \end{matrix}$

Again, by Assumption 1.5, we conclude that $w = \partial_{t} v \equiv 0$ , and, therefore, returning to (3.21) we deduce that $v \equiv 0$ .

Suppose that $α = 0$ . Thus, since $f \in C^{2} (R)$ from Taylor’s formula there exists $s_{0} \in (0, s)$ such that $\begin{matrix} (3.23) & f (s) = f^{'} (0) s + \frac{1}{2} f^{″} (s_{0}) s^{2} . \end{matrix}$ Note that we can assume without loss of generality that $s > 0$ . Indeed, for $s = 0$ equation (3.23) holds trivially and for the case $s < 0$ we consider $g : [- s, 0] \to R$ given by $g (t) = f (- t)$ with $s > 0$ . Define $\begin{matrix} (3.24) & R_{f} (s) = \frac{1}{2} f^{″} (s_{0}) s^{2} . \end{matrix}$ Thus, from (1.3) we obtain $\begin{array}{rcl} | R_{f} (s) | & = & \frac{1}{2} | f^{″} (s_{0}) | s^{2} \\ ≲ & {(1 + | s_{0} |)}^{p - 2} s^{2} \\ = & \frac{{(1 + | s_{0} |)}^{p - 1}}{(1 + | s_{0} |)} s^{2} \\ ≲ & {(1 + | s_{0} |)}^{p - 1} s^{2} \\ (3.25) & ≲ & | s |^{2} + | s |^{p + 1} for all s \in R . \end{array}$

Therefore, $\begin{matrix} (3.26) & \frac{1}{α_{m}} f (α_{m} v^{m}) = f^{'} (0) v^{m} + \frac{R_{f} (α_{m} v^{m})}{α_{m}} \end{matrix}$ and $\begin{matrix} (3.27) & | \frac{R_{f} (α_{m} v^{m})}{α_{m}} | ≲ α_{m} {| v^{m} |}^{2} + | α_{m} |^{p} {| v^{m} |}^{p + 1} if 1 ⩽ p < \frac{n}{n - 2} . \end{matrix}$ Now, we are going to prove that $\begin{matrix} \frac{R_{f} (α_{m} v^{m})}{α_{m}} ⇀ 0 weakly in L^{2} (0, T; L^{2} (Ω)) as m \to + \infty . \end{matrix}$ Indeed, observe that $R_{f} (s) = f (s) - f^{'} (0) s$ , then taking (1.4) into account we obtain $\begin{array}{rcl} {‖ \frac{R_{f} (α_{m} v^{m})}{α_{m}} ‖}_{L^{2} (0, T; L^{2} (Ω))}^{2} & ≲ & \int_{0}^{T} \int_{Ω} {| \frac{f (α_{m} v^{m})}{α_{m}} |}^{2} d x d t + \int_{0}^{T} \int_{Ω} {| f^{'} (0) v^{m} |}^{2} d x d t \\ ≲ & \frac{1}{α_{m}^{2}} \int_{0}^{T} \int_{Ω} {(| α_{m} v^{m} | + {| α_{m} v^{m} |}^{p})}^{2} d x d t + \int_{0}^{T} \int_{Ω} {| v^{m} |}^{2} d x d t \\ (3.28) & ≲ & {‖ v^{m} ‖}_{L^{2} (0, T; L^{2} (Ω))}^{2} + {(α_{m})}^{2 (p - 1)} {‖ v^{m} ‖}_{L^{2 p} (0, T; L^{2 p} (Ω))}^{2 p} . \end{array}$ Using Sobolev’s embeddings, the boundedness of the sequence ${v^{m}}_{m \in N}$ and that $p ⩾ 1$ , we obtain that there exists $C > 0$ verifying $\begin{matrix} (3.29) & {‖ \frac{R_{f} (α_{m} v^{m})}{α_{m}} ‖}_{L^{2} (0, T; L^{2} (Ω))} ⩽ C \end{matrix}$ and, consequently, there is $r \in L^{2} (0, T; L^{2} (Ω))$ such that $\begin{matrix} (3.30) & \frac{R_{f} (α_{m} v^{m})}{α_{m}} ⇀ r weakly in L^{2} (0, T; L^{2} (Ω)) . \end{matrix}$ On the other hand, using (3.27) we have $\begin{matrix} (3.31) & {‖ \frac{R_{f} (α_{m} v^{m})}{α_{m}} ‖}_{L^{1} (0, T; L^{1} (Ω))}^{2} ≲ α_{m} {‖ v^{m} ‖}_{L^{2} (0, T; L^{2} (Ω))}^{2} + {(α_{m})}^{p} {‖ v^{m} ‖}_{L^{p + 1} (0, T; L^{p + 1} (Ω))}^{p + 1} \to 0, \end{matrix}$ as $m \to \infty$ . From (3.30) and (3.31) and the embedding $L^{2} (0, T; L^{2} (Ω)) ↪ L^{1} (0, T; L^{1} (Ω))$ we obtain $\begin{matrix} (3.32) & \frac{R_{f} (α_{m} v^{m})}{α_{m}} ⇀ 0 weakly in L^{2} (0, T; L^{2} (Ω)) . \end{matrix}$ Thus, from (3.26), (3.32) we obtain $\begin{matrix} (3.33) & \frac{1}{α_{m}} f (α_{m} v^{m}) ⇀ f^{'} (0) v weakly in L^{2} (0, T; L^{2} (Ω)) . \end{matrix}$

Passing to the limit in (3.14) as $m \to + \infty$ , we obtain $\begin{matrix} (3.34) & \{\begin{matrix} ρ (x) \partial_{t}^{2} v - div (K (x) \nabla (v)) + f^{'} (0) v = 0 & in Ω \times (0, + \infty), \\ v = 0 & on \partial Ω \times (0, + \infty), \\ \partial_{t} v = 0 & in Ω ∖ C, \end{matrix} \end{matrix}$ and defining $w = \partial_{t} v$ , it satisfies the following problem: $\begin{matrix} (3.35) & \{\begin{matrix} ρ (x) \partial_{t}^{2} w - div (K (x) \nabla (w)) + f^{'} (0) w = 0 & in Ω \times (0, + \infty), \\ w = 0 & on \partial Ω \times (0, + \infty), \\ w = 0 & in Ω ∖ C, \end{matrix} \end{matrix}$

By employing Assumption 1.5 again we obtain that $w = \partial_{t} v \equiv 0$ and returning to (3.34) we deduce that $v \equiv 0$ .

Then, in both cases $α = 0$ and $α > 0$ , we obtain that $v \equiv 0$ . Taking (3.19), (3.26) and (3.28) into account it is easy to see that $\begin{matrix} (3.36) & \frac{1}{α_{m}} f (α_{m} v^{m}) \to 0 in L^{2} (0, T, L^{2} (Ω)) . \end{matrix}$

Now, let’s consider $□ = ρ (x) \partial_{t}^{2} - div (K (x) \nabla (\cdot))$ the d’Alembert operator. Thanks to (3.14), we have that $\begin{matrix} □ v^{m} = - \frac{1}{α_{m}} f (u^{m}) + div (a (x) \nabla \partial_{t} v^{m}) - b (x) \partial_{t} v^{m}, \end{matrix}$ that is, $\begin{matrix} (3.37) & \partial_{t} □ v^{m} = \partial_{t} (- \frac{1}{α_{m}} f (u^{m}) + div (a (x) \nabla \partial_{t} v^{m}) - b (x) \partial_{t} v^{m}) . \end{matrix}$

Let us analyse the terms on the right hand side of (3.37).

Analysis of $I_{1}^{m} : = - \frac{1}{α_{m}} f (u^{m})$ .

Employing convergence (3.36), we deduce that $I_{1}^{m} \to 0$ strongly in $L^{2} (Ω \times (0, T))$ as $m \to + \infty$ .

Analysis of $I_{3}^{m} : = - b (x) \partial_{t} v^{m}$ .

We trivially obtain from (3.13) that $- b (x) \partial_{t} v^{m} \to 0$ strongly in $L^{2} (Ω \times (0, T))$ as $m \to + \infty$ .

Analysis of $I_{2}^{m} : = div (a (x) \nabla \partial_{t} v^{m})$ .

Recalling convergence (3.13), we deduce that $a (x) \nabla \partial_{t} v^{m} \to 0 strongly in L^{2} (Ω \times (0, T))$ as $m \to + \infty$ and, consequently, $div (a (x) \nabla \partial_{t} v^{m}) \to 0 strongly in H_{loc}^{- 1} (Ω \times (0, T))$ as $m \to + \infty$ .

The above convergences yield $\begin{matrix} \partial_{t} (- \frac{1}{α_{m}} f (u^{m}) + div (a (x) \nabla \partial_{t} v^{m}) - b (x) \partial_{t} v^{m}) \to 0 strongly in H_{loc}^{- 2} (Ω \times (0, T)), \end{matrix}$ that is, from (3.37) we conclude that $\begin{matrix} (3.38) & □ \partial_{t} v^{m} \to 0 strongly in H_{loc}^{- 2} (Ω \times (0, T)), \end{matrix}$ as $m \to + \infty$ .

Let μ be the microlocal defect measure associated with ${\partial_{t} v^{m}}$ in $L_{loc}^{2} (Ω \times (0, T))$ . The convergence (3.38) guarantees that the $supp (μ)$ is contained in the characteristic set of the wave operator (see Gérard [15] (Proposition 2.1 and Corollary 2.2) or Theorem A.2 in the Appendix).

On the other hand, Poincaré inequality combined with the existence of the dissipative effects, yield $\begin{matrix} (3.39) & \partial_{t} v^{m} \to 0 in (Ω ∖ C) \times (0, T) . \end{matrix}$

Our goal is to propagate the convergence of $\partial_{t} v^{m}$ to zero from $L^{2} ((Ω ∖ C) \times (0, T))$ to the whole space $L^{2} (Ω \times (0, T))$ . However, the convergence in (3.38) is not regular enough to guarantee the desired propagation. This lack of regularity comes from the following convergence: $\begin{matrix} \partial_{t} (div (a (x) \nabla \partial_{t} u^{m})) \to 0 strongly in H_{loc}^{- 2} (Ω \times (0, T)), as m \to + \infty . \end{matrix}$

At this point, the presence of the frictional damping in the neighbourhood $V_{ε}$ of the boundary $\partial A$ of $A : = {x \in Ω : a (x) = 0}$ plays an essential role to obtain the propagation to the whole space. Indeed, note that $a (x) \nabla \partial_{t} u^{m} = 0$ in $A \times (0, T)$ , consequently, from (3.37) we have that $\begin{matrix} (3.40) & □ \partial_{t} v^{m} = \partial_{t} (- \frac{1}{α_{m}} f (u^{m}) - \partial_{t} v^{m}) \to 0 in H_{loc}^{- 1} (int (A) \times (0, T)), \end{matrix}$ as $m \to + \infty$ .

Furthermore, from Proposition A.1 and Theorem A.4 found in the Appendix, we deduce that $supp (μ)$ in $(int (A) \times (0, T)) \times S^{d}$ is a union of curves like $\begin{matrix} (3.41) & t \in I \mapsto m_{\pm} (t) = (t, x (t), \frac{\pm 1}{\sqrt{1 + | G (x) \dot{x} |^{2}}}, \frac{\mp G (x) \dot{x}}{\sqrt{1 + | G (x) \dot{x} |^{2}}}), \end{matrix}$ where $t \in I \mapsto x (t) \in Ω$ is a geodesic associated to the metric.

The above convergence yields that μ propagates along the bicharacteristic flow of the operator $□ (\cdot)$ , which means that if there is $ω_{0} = (t_{0}, x_{0}, τ_{0}, ξ_{0})$ such that $ω_{0} = (t_{0}, x_{0}, τ_{0}, ξ_{0}) \notin supp (μ)$ , the whole bicharacteristic issued from $ω_{0}$ does not belong to $supp (μ)$ (see Proposition A.1 and Theorem A.4 in the Appendix). However, since $supp (μ) \subset C \times (0, T) \subset A \times (0, T)$ , and the frictional damping acts in both sides of the boundary $\partial A$ , we can propagate the kinetic energy from $(V_{ε / 2} \cap A) \times (0, T)$ towards the set $C \times (0, T)$ . Indeed, let $t_{0} \in (0, + \infty)$ and let x be a geodesic of G defined near $t_{0}$ . Once the geodesics inside $C$ , enter necessarily in the region ω, they also intersect the set $A ∖ C$ , then, for any geodesic of the metric G, with $0 \in I$ there exists $t > 0$ such that $m_{\pm} (t)$ does not belong to the $supp (μ)$ , so that $m_{\pm} (t_{0})$ does not belong as well. Therefore, $supp (μ)$ is empty. As a consequence, $\partial_{t} v^{m} \to 0$ in $L_{loc}^{2} (Ω \times (0, T))$ from Remark A.1. Since, $\partial_{t} v^{m} \to 0$ in $L^{2} (Ω ∖ C \times (0, T))$ we obtain $\begin{matrix} (3.42) & \int_{0}^{T} \int_{Ω} {| \partial_{t} v^{m} (x, t) |}^{2} d x d t \to 0, as m \to + \infty . \end{matrix}$

Then, we prove that $E_{v^{m}} (0) \to 0$ as $m \to + \infty$ . In fact, consider the following cut-off function $\begin{matrix} θ \in C^{\infty} (0, T), 0 ⩽ θ (t) ⩽ 1, θ (t) = 1 in (ε, T - ε), ε > 0 . \end{matrix}$

Multiplying equation (3.14) by $v^{m} θ$ and integrating by parts, we infer $\begin{array}{l} - \int_{0}^{T} θ (t) \int_{Ω} ρ (x) {| v_{t}^{m} |}^{2} d x d t - \int_{0}^{T} θ^{'} (t) \int_{Ω} ρ (x) v_{t}^{m} v^{m} d x d t \\ + \int_{0}^{T} θ (t) \int_{Ω} {(\nabla v^{m})}^{⊤} \cdot K (x) \cdot \nabla v^{m} d x d t + \frac{1}{α_{m}} \int_{0}^{T} θ (t) \int_{Ω} f (α_{m} v^{m}) v^{m} d x d t \\ (3.43) & + \int_{0}^{T} θ (t) \int_{Ω} a (x) \nabla v_{t}^{m} \cdot \nabla v^{m} d x d t + \int_{0}^{T} θ (t) \int_{Ω} b (x) \partial_{t} v^{m} v^{m} d x d t = 0 . \end{array}$

Considering the convergences (3.13), (3.17)–(3.19) and (3.42) and employing the fact that $v = 0$ , identity (3.43) yields $\begin{matrix} (3.44) & lim_{m \to + \infty} \int_{ε}^{T - ε} \int_{Ω} {(\nabla v^{m})}^{⊤} \cdot K (x) \cdot \nabla v^{m} + \frac{1}{α_{m}} f (α_{m} v^{m}) v^{m} d x d t = 0 . \end{matrix}$

In addition, using the definition of functions f and F and taking advantage of property (1.6), the last convergence implies $\begin{matrix} (3.45) & lim_{m \to + \infty} \frac{1}{α_{m}^{2}} \int_{ε}^{T - ε} \int_{Ω} F (u^{m}) d x d t = 0 . \end{matrix}$

Convergences (3.42), (3.44) and (3.45) establish that $\int_{ε}^{T - ε} E_{v^{m}} (t) \to 0$ and recalling that the energy functional is a non-increasing function, we obtain $\begin{matrix} (3.46) & (T - 2 ε) E_{v^{m}} (T - ε) \to 0, as m \to + \infty . \end{matrix}$

Using the energy identity we deduce that $\begin{matrix} E_{v^{m}} (T - ε) - E_{v^{m}} (ε) = - \int_{ε}^{T - ε} \int_{Ω} a (x) {| \nabla \partial_{t} v^{m} |}^{2} + b (x) {| \partial_{t} v^{m} |}^{2} d x d t, \end{matrix}$ which, in view of convergences (3.13) and (3.46) and the arbitrariness of $ε > 0$ , implies that $E_{v^{m}} (0) \to 0$ as $m \to + \infty$ .

Then, according to (3.16), the desired contradiction is achieved and we finish the proof. □

In what follows, we are going to conclude the exponential stability to the problem (1.1).

Thanks to inequality (3.2), the energy identity (1.8) and the fact that the map $t \mapsto E_{u} (t)$ is a non-increasing function, we obtain $\begin{matrix} (3.47) & \begin{matrix} E_{u} (T_{0}) & ⩽ C \int_{0}^{T_{0}} \int_{Ω} (a (x) | \nabla \partial_{t} u |^{2} + b (x) | \partial_{t} u |^{2}) d x d t \\ = C (E_{u} (0) - E_{u} (T_{0})), \end{matrix} \end{matrix}$ that is, $\begin{matrix} (3.48) & E_{u} (T_{0}) ⩽ (\frac{C}{1 + C}) E_{u} (0) . \end{matrix}$

Repeating the same steps for $m T_{0}$ , $m \in N$ , $m ⩾ 1$ , we deduce $\begin{matrix} E_{u} (m T_{0}) ⩽ \frac{1}{{(1 + \hat{C})}^{m}} E_{u} (0), \end{matrix}$ where $\hat{C} = C^{- 1}$ . Consider $t ⩾ T_{0}$ and $t = m T_{0} + r$ , $0 ⩽ r < T_{0}$ . Thus, $\begin{matrix} E_{u} (t) ⩽ E_{u} (t - r) = E_{u} (m T_{0}) ⩽ \frac{1}{{(1 + \hat{C})}^{m}} E_{u} (0) = \frac{1}{{(1 + \hat{C})}^{\frac{t - r}{T_{0}}}} E_{u} (0) . \end{matrix}$

Defining $C : = e^{\frac{r}{T_{0}} ln (1 + \hat{C})}$ and $λ_{0} : = \frac{ln (1 + \hat{C})}{T_{0}} > 0$ , we obtain $\begin{matrix} (3.49) & E_{u} (t) ⩽ C e^{- λ_{0} t} E_{u} (0) for all t ⩾ T_{0}, \end{matrix}$ which proves the exponential decay to problem (1.1) and we prove the following result:

Theorem 3.1.

Under the Assumptions 1.1 , 1.3 , 1.4 and 1.5 , there exist positive constants C and γ such that the following exponential decay holds $\begin{matrix} (3.50) & E_{u} (t) ⩽ C e^{- γ t} E_{u} (0), for all t ⩾ T_{0}, \end{matrix}$ for every solution to problem ( 1.1 ), provided that the initial data are taken in bounded sets of the phase-space $H : = H_{0}^{1} (Ω) \times L^{2} (Ω)$ .

Footnotes

Acknowledgements

Research of Marcelo M. Cavalcanti is partially supported by the CNPq Grant 300631/2003-0.

Research of Victor Hugo Gonzalez Martinez is supported by CAPES.

Appendix

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Exponential decay for the semilinear wave equation with localized frictional and Kelvin–Voigt dissipating mechanisms

Abstract

Keywords

1. Introduction

1.1. Description of the problem

2. Well-posedness

Theorem 2.1 (Global Well-Posedness).

Footnotes

Acknowledgements

Appendix

References