Indirect Stabilization of Nonlinear Coupled Wave Equations in the Presence of Time Delay or Indefinite Damping

Abstract

This paper explores the exponential stability of two nonlinear wave equations coupled through their velocities. The analysis is divided into two main cases. First, we consider a system where one equation is damped, while the other experiences a discrete time delay. By reformulating the problem in an abstract framework, we use semigroup theory and energy methods to establish well-posedness and derive conditions that guarantee exponential energy decay. In the second case, we examine a scenario where a frictional damping term appears in the first equation, while the second equation contains an indefinite damping term, namely with a sign-changing coefficient. Although this setup can be viewed as a special case of the first, we analyze it separately and show that exponential stability still holds under a weaker condition.

Keywords

stabilization nonlinear hyperbolic systems delay feedback indefinite damping

1. Introduction

Time delays are an inherent feature of numerous real-world systems, particularly those where the current state is influenced by historical values. Consequently, their inclusion in mathematical models is essential for accurately representing phenomena in biology, mechanics, and engineering.

Over the past decades, substantial research has focused on the stability of systems with delayed damping, a phenomenon often stemming from sources such as control force computation, signal transmission, or measurement processing.

It is well-established that time delays frequently induce instability in systems that are otherwise, that is without any time lags, uniformly asymptotically stable (see e.g. Datko, 1991; Nicaise & Pignotti, 2006). This destabilizing effect has consequently motivated significant research efforts aimed at developing effective strategies to mitigate or control these delay effects; see for instance (Adimy & Crauste, 2003; Ammari et al., 2010, 2022; Continelli & Pignotti, 2025; Nicaise & Pignotti, 2006; Pignotti, 2012; Xu et al., 2006).

In many practical applications, system dynamics are too complex to be described by a single scalar equation, often requiring coupled systems of equations instead. Extensive literature exists on the stabilization of coupled wave equations without delay (Alabau-Boussouira, 2002, 2003, 2013; Alabau-Boussouira & Léautaud, 2013; Alabau-Boussouira et al., 2017; Avdonin et al., 2013; Bennour et al., 2017; Gerbi et al., 2021; Koumaiha et al., 2022; Liu & Rao, 2009; Mokhtari & Ammar Khodja, 2022; Wehbe & Youssef, 2011); however, the impact of delays on such coupled systems remains less explored. For weakly coupled systems, some theoretical results have been achieved in Oliveira and Oquendo (2020), Rebiai and Sidi Ali (2014), and Xu and Zhang (2020), while recent progress for strongly coupled systems can be found in Akil, Badawi, Nicaise, et al. (2021), Akil, Badawi, Wehbe (2021), Ayechi et al. (2025), Moumni et al. (2024), and Silga et al. (2022).

Let $Ω$ be a bounded open domain in $R^{n}$ with a boundary $\partial Ω$ of class $C^{2}$ . In Gerbi et al. (2021), the authors have investigated the energy decay of a system of two linear wave equations coupled through velocities described by (see also Alabau-Boussouira et al., 2017 for the case of a nonlinear damping):

{\begin{cases} u_{t t} (x, t) - Δ u (x, t) + b (x) y_{t} (x, t) + a (x) u_{t} (x, t) = 0, & in Ω \times (0, \infty), \\ y_{t t} (x, t) - Δ y (x, t) - b (x) u_{t} (x, t) = 0, & in Ω \times (0, \infty), \\ u (x, t) = y (x, t) = 0, & on \partial Ω \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & in Ω, \\ y (x, 0) = y_{0} (x), y_{t} (x, 0) = y_{1} (x), & in Ω . \end{cases}

(1.1)

where

a, b \in C^{0} (\bar{Ω})

and

a

is a function satisfying

a (x) \geq 0 almost everywhere (a.e.) in Ω,

and for a given open subset

ω \subset Ω

a (x) > a_{0} > 0 a.e. in ω .

Moreover, it is imposed that there exists a non-empty open subset

ω_{b}

Ω

such that

\bar{ω_{b}} \subset {x \in Ω : b (x) \neq 0},

(1.2)

where

\bar{ω_{b}}

is the closure of

ω_{b}

In order to analyze the energy decay properties of system (1.1), the following assumptions have been considered:

Hypothesis 1.1

The coupling region $ω_{b}$ is contained within the damping region $ω$ $(ω_{b} \subset ω)$ , and $ω_{b}$ satisfies the so-called Geometric Control Condition (GCC), introduced in Bardos et al. (1992) and recalled below in Definition 2.1.

Hypothesis 1.2

The coupling coefficient $b (x)$ is small and positive, satisfying $0 \leq b (x) \leq b_{+}$ , $\forall x \in Ω$ , where $b_{+} \in (0, b^{*}]$ and $b^{*}$ is a constant depending on the domain and the control region. Additionally, there exists $b_{-} > 0$ such that $b (x) > b_{-}$ almost everywhere in $ω_{b}$ . Moreover, both the coupling region $ω_{b}$ and the damping region $ω$ satisfy an appropriate geometric condition known as the Piecewise Multipliers Geometric Condition (PMGC), introduced in Liu (1997), applied in Alabau-Boussouira et al. (2017) and recalled below in Definition 2.1.

Under Assumption 1.1, the authors in Gerbi et al. (2021) combined a frequency domain approach with the multiplier technique to establish the exponential decay of the corresponding energy for system (1.1). On the other hand, under Assumption 1.2, the authors in Alabau-Boussouira et al. (2017) employed multiplier techniques to achieve similar exponential stability results.

In this work, we aim to extend the results in Alabau-Boussouira et al. (2017), and Gerbi et al. (2021) by considering a more complex setting where the system incorporates a time delay in one equation and nonlinear source terms. Specifically, we investigate the following coupled wave system:

{\begin{cases} u_{t t} (x, t) - Δ u (x, t) + b (x) y_{t} (x, t) + a_{1} (x) u_{t} (x, t) = f_{1} (u (x, t)), & in Ω \times (0, \infty), \\ y_{t t} (x, t) - Δ y (x, t) - b (x) u_{t} (x, t) + a_{2} (x) y_{t} (x, t - τ) = f_{2} (y (x, t)), & in Ω \times (0, \infty), \\ u (x, t) = y (x, t) = 0, & on \partial Ω \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & in Ω, \\ y (x, 0) = y_{0} (x), y_{t} (x, 0) = y_{1} (x), & in Ω, \\ y_{t} (x, t) = g (x, t), & in Ω \times [- τ, 0] . \end{cases}

(1.3)

Here, the functions

a_{1} \in C^{0} (\bar{Ω})

and

a_{2} \in L^{\infty} (Ω)

are given coefficients. The coefficient

a_{1}

satisfies the conditions:

a_{1} (x) \geq 0 a.e in Ω,

and

a_{1} (x) > a_{0} > 0 a.e in ω \subset Ω .

The initial state $(u_{0}, u_{1}, y_{0}, y_{1}, g)$ belongs to an appropriate function space, which will be specified later. Moreover, $f_{1}$ and $f_{2}$ are two nonlinear functions satisfying suitable hypotheses that will be specified in the next sections. Finally, $τ > 0$ represents the constant time delay.

Unlike previous studies that considered coupled wave systems where both the damping and delay terms appear in the same equation (Akil, Badawi, Wehbe, 2021; Moumni et al., 2024; Oliveira & Oquendo, 2020; Silga et al., 2022), our focus is on the more challenging scenario where the damping and delay are distributed across different equations. This configuration complicates the stability analysis. Indeed, in the classical setting where both mechanisms are in the same equation, the damping can directly counterbalance the delay effects (cf. Nicaise & Pignotti, 2006). In our case, this interaction is not straightforward, leading to additional mathematical challenges. This motivates our investigation into establishing new energy decay results for system (1.3) under appropriate conditions.

Furthermore, we include nonlinear terms in the system, taking into account the presence of source terms. Nonlinear effects are often present in delay differential equations that come from real-life applications. Various works have studied hyperbolic equations with suitable nonlinear source terms. For example, in Ayechi et al. (2025), Chellaoua et al. (2023), Fragnelli and Pignotti (2016), Paolucci and Pignotti (2022), and Pignotti (2024), the authors study the stability of solutions for some abstract wave equations with nonlinear terms satisfying some local Lipschitz continuity assumption. The presence of nonlinear terms makes the models more difficult to deal with and the stability issues require a more sophisticated analysis.

In practical situations, such as in elasticity and viscoelasticity, these nonlinearities naturally appear and lead to nonlinear source terms. In Alabau-Boussouira et al. (2008), Berrimi and Messaoudi (2006), and Luo and Xiao (2023), the authors prove energy decay estimates for such systems.

To the best of our knowledge, this is the first work in which the destabilizing effect of a delay feedback present in one equation is compensated by means of a standard frictional damping in the other equation. Moreover, the source terms introduce stability issues too. This combination makes the mathematical analysis more delicate.

Our goal is, then, to explore how the interaction between delay, nonlinearity, damping, and coupling mechanisms influences the stability properties of the system, and to derive sufficient conditions ensuring exponential energy decay.

Additionally, we also examine the particular case $τ = 0$ , which corresponds to a system with indefinite damping:

{\begin{cases} u_{t t} (x, t) - Δ u (x, t) + b (x) y_{t} (x, t) + a_{1} (x) u_{t} (x, t) = f_{1} (u (x, t)), & in Ω \times (0, \infty), \\ y_{t t} (x, t) - Δ y (x, t) - b (x) u_{t} (x, t) + a_{2} (x) y_{t} (x, t) = f_{2} (y (x, t)), & in Ω \times (0, \infty), \\ u (x, t) = y (x, t) = 0, & on \partial Ω \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & in Ω, \\ y (x, 0) = y_{0} (x), y_{t} (x, 0) = y_{1} (x), & in Ω . \end{cases}

(1.4)

This system represents a limiting case of our study but is considered separately since we can obtain the result under weaker conditions when

τ = 0

. In particular, we establish exponential stability for (1.4) under a suitable assumption on

a_{2}^{-} (x) = - min {a_{2} (x), 0}

. The study of indefinite damping functions in scalar wave equations has been considered in previous works, such as (Benaddi & Rao, 2000; Freitas & Zuazua, 1996; Haraux, 1989; López-Gómez, 1997; Menz, 2007) but for coupled wave equations has not been done as far as we know.

2. The Delayed Coupled System

In this section, we analyze the well-posedness and stability of the nonlinear delayed coupled system (1.3). Before presenting our main result of this section, we recall the GCC introduced by Bardos et al. (1992) and the PMGC introduced by Liu (1997).

Definition 2.1
We say that a subset $ω$ of $Ω$ satisfies the GCC if there exists a time $T > 0$ such that every ray of the geometrical optics starting at any point $x \in Ω$ at $t = 0$ enters the region $ω$ within time $T$ .
Definition 2.2
We say that a subset $ω \subset Ω$ satisfies the $P M G C$ , if there exist subsets $Ω_{j} \subset Ω$ with Lipschitz boundaries and points $x_{j} \in R^{N}, j = 1, \dots, L$ , such that $Ω_{i} \cap Ω_{j} = \emptyset$ for $i \neq j$ and $ω$ contains a neighborhood in $Ω$ of the set $\cup_{j = 1}^{L} γ_{j} (x_{j}) \cup (Ω ∖ \cup_{j = 1}^{L} Ω_{j})$ , where $γ_{j} (x_{j}) = {x \in \partial Ω_{j} : (x - x_{j}) \cdot ν_{j} (x) ⩾ 0}$ and $ν_{j}$ is the outward unit normal to $\partial Ω_{j}$ .

We introduce the energy space
$H = {(H_{0}^{1} (Ω) \times L^{2} (Ω))}^{2} .$
Considering real-valued solutions, $H$ is defined as a real Hilbert space equipped with the inner product:
$(U, \tilde{U})_{H} = \int_{Ω} \nabla u_{1} \cdot \nabla {\tilde{u}}_{1} d x + \int_{Ω} u_{2} {\tilde{u}}_{2} d x + \int_{Ω} \nabla y_{1} \cdot \nabla {\tilde{y}}_{1} d x + \int_{Ω} y_{2} {\tilde{y}}_{2} d x,$
for all $U = (u_{1}, u_{2}, y_{1}, y_{2})$ and $\tilde{U} = ({\tilde{u}}_{1}, {\tilde{u}}_{2}, {\tilde{y}}_{1}, {\tilde{y}}_{2})$ in $H$ . The associated norm is given by
$‖ U ‖_{H} = \sqrt{(U, U)_{H}} .$

Setting $u_{1} = u$ , $u_{2} = u_{t}$ , $y_{1} = y$ , and $y_{2} = y_{t}$ , we define the state variable
$U = [\begin{matrix} u_{1} \\ u_{2} \\ y_{1} \\ y_{2} \end{matrix}], U_{0} = [\begin{matrix} u_{0} \\ u_{1} \\ y_{0} \\ y_{1} \end{matrix}] .$

We also introduce
$Ψ (s) = [\begin{matrix} 0 \\ 0 \\ 0 \\ a_{2} g (s) \end{matrix}], B U (t) = [\begin{matrix} 0 \\ 0 \\ 0 \\ a_{2} y_{2} (t) \end{matrix}], F (U (t)) = [\begin{matrix} 0 \\ f_{1} (u (x, t)) \\ 0 \\ f_{2} (y (x, t)) \end{matrix}] .$

Next, we define the linear operator $A : D (A) \subset H \to H$ by
$D (A) = {((H^{2} (Ω) \cap H_{0}^{1} (Ω)) \times H_{0}^{1} (Ω))}^{2}, A U = [\begin{matrix} u_{2} \\ Δ u_{1} - b y_{2} - a_{1} u_{2} \\ y_{2} \\ Δ y_{1} + b u_{2} \end{matrix}] .$

It is well known that the operator $A$ is non-negative and generates an exponentially stable contraction semigroup $(S (t))_{t \geq 0}$ on $H$ (see Alabau-Boussouira et al., 2017, Remark 3.3 and Gerbi et al., 2021, Theorem 3.4) under either Assumption 1.1 or Assumption 1.2. More precisely, there exist constants $M > 0$ and $α > 0$ such that
$‖ S (t) ‖_{L (H)} \leq M e^{- α t}, \forall t \geq 0.$
(2.1)

Thus, the delayed coupled system (1.3) can be rewritten in the following abstract form:
${\begin{cases} \dot{U} (t) = A U (t) - B U (t - τ) + F (U (t)), & t > 0, \\ U (0) = U_{0}, \\ B U (s) = Ψ (s), & s \in [- τ, 0] . \end{cases}$
(2.2)

In order to study well-posedness and the exponential stability of system (2.2), we make the following assumptions on $f_{1}$ and $f_{2}$ :
Hypothesis 2.1
$f_{1}, f_{2}$ are continuous functions such that
$f_{1} (0) = f_{2} (0) = 0$ ;

For all $r > 0$ there exists a constant $L (r) > 0$ such that, for all $u, v \in H_{0}^{1} (Ω)$ satisfying $‖ \nabla u ‖_{L^{2} (Ω)} \leq r$ and $‖ \nabla v ‖_{L^{2} (Ω)} \leq r$ , one has
$‖ f_{i} (u) - f_{i} (v) ‖_{L^{2} (Ω)} \leq L (r) {‖ \nabla u - \nabla v ‖}_{L^{2} (Ω)}$
for $i = 1, 2$ ; and $L (r) \to 0$ as $r \to 0.$

There exist two strictly increasing continuous functions $h_{1}, h_{2} : R_{+} \to R_{+},$ $h_{1} (0) = h_{2} (0) = 0,$ such that
$| \int_{Ω} f_{i} (u) u d x | \leq h_{i} ({‖ \nabla u ‖}_{L^{2} (Ω)}) {‖ \nabla u ‖}_{L^{2} (Ω)}^{2}$
(2.3)
for all $u \in H_{0}^{1} (Ω)$ and $i = 1, 2$ .

Furthermore, thanks to Hypothesis 2.1, we have that $F (0) = 0$ and for any $r > 0$ there exists a constant $L (r) > 0$ such that
$‖ F (U) - F (V) ‖_{H} \leq L (r) ‖ U - V ‖_{H}$
whenever $‖ U ‖_{H} \leq r$ and $‖ V ‖_{H} \leq r$ . In particular,
$‖ F (U) ‖_{H} \leq L (r) ‖ U ‖_{H} .$
Let us define
$F_{i} (s) := \int_{0}^{s} f_{i} (r) d r, \forall s \in R,$
(2.4)
for $i = 1, 2$ .

Under Hypothesis 2.1, we derive the following estimates for the nonlinear terms $\int_{Ω} F_{1} (u (x)) d x$ and $\int_{Ω} F_{2} (u (x)) d x$ .
Lemma 2.1
Assume Hypothesis 2.1 holds. Then, for all $u \in H_{0}^{1} (Ω)$ and $i = 1, 2$ , we have
$| \int_{Ω} F_{i} (u (x)) d x | \leq \frac{1}{2} h_{i} (‖ \nabla u ‖_{L^{2} (Ω)}) ‖ \nabla u ‖_{L^{2} (Ω)}^{2} .$
(2.5)
Proof.
Let $u \in H_{0}^{1} (Ω)$ and $i \in {1, 2}$ . Using the identity
$\frac{d}{d s} F_{i} (s u) = F_{i}^{'} (s u) u = f_{i} (s u) u,$
we obtain
$\int_{Ω} F_{i} (u (x)) d x = \int_{Ω} \int_{0}^{1} f_{i} (s u) u d s d x = \int_{0}^{1} \int_{Ω} f_{i} (s u) s u d x \frac{d s}{s} .$
Applying (2.3), we deduce
$\begin{aligned} | \int_{Ω} F_{i} (u (x)) d x | & \leq \int_{0}^{1} h_{i} (‖ \nabla (s u) ‖_{L^{2} (Ω)}) ‖ \nabla (s u) ‖_{L^{2} (Ω)}^{2} \frac{d s}{s} \\ \leq \int_{0}^{1} h_{i} (‖ \nabla (s u) ‖_{L^{2} (Ω)}) s^{2} ‖ \nabla u ‖_{L^{2} (Ω)}^{2} \frac{d s}{s} \\ \leq \frac{1}{2} h_{i} (‖ \nabla u ‖_{L^{2} (Ω)}) ‖ \nabla u ‖_{L^{2} (Ω)}^{2} . \end{aligned}$

We begin by establishing the local existence and uniqueness of solutions, as stated in the following lemma.
Lemma 2.2
Let us consider the system (2.2) with initial data $U_{0} \in H$ and $Ψ \in C ([- τ, 0]; H) .$ Then, there exists a unique continuous local solution $U$ defined on a time interval $[0, δ)$ , for some $δ > 0$ .
Proof.
Note that for $t \in (0, τ),$ $t - τ \in (- τ, 0) .$ Then, for $t \in (0, τ),$ we have $B U (t - τ) = Ψ (t - τ) = a_{2} g (t - τ)$ and we can rewrite the abstract system (2.2) as an non-delayed problem:
${\begin{cases} \dot{U} (t) = A U (t) - a_{2} g (t - τ) + F (U (t)), & t \in (0, τ), \\ U (0) = U_{0} . \end{cases}$
Then, we can apply the classical theory of nonlinear semigroups (Pazy, 2012) obtaining the existence of a unique solution on a set $[0, δ)$ , with $δ \leq τ$ .

Observe that, for solution to (2.2), the following Duhamel formula holds:
$U (t) = S (t) U_{0} + \int_{0}^{t} S (t - s) B U (s - τ) d s + \int_{0}^{t} S (t - s) F (U (s)) d s .$
(2.6)
In the spirit of Paolucci and Pignotti (2022), in order to have the existence and uniqueness of a global solution of the system (2.2), we introduce the following energy functional associated to system (1.3):
Definition 2.3
Let $(u, y)$ be a mild solution of (1.3) and define its energy by
$\begin{aligned} E (t) & = \frac{1}{2} \int_{Ω} {u_{t}^{2} (x, t) + | \nabla u (x, t) |^{2} + y_{t}^{2} (x, t) + | \nabla y (x, t) |^{2}} d x \\ - \int_{Ω} F_{1} (u (x, t)) d x - \int_{Ω} F_{2} (y (x, t)) d x + \frac{1}{2} \int_{Ω} \int_{t - τ}^{t} | a_{2} (x) | y_{t}^{2} (x, s) d s d x . \end{aligned}$
(2.7)

We can deduce a first estimate for small enough initial data.
Lemma 2.3
Assume Hypothesis 2.1. Let $U = (u, u_{t}, y, y_{t})$ be a non-trivial solution of (2.2) defined on an interval $[0, T),$ for some $T > 0.$ If $h_{1} (‖ \nabla u_{0} ‖_{L^{2} (Ω)}) < \frac{1}{2}$ and $h_{2} (‖ \nabla y_{0} ‖_{L^{2} (Ω)}) < \frac{1}{2}$ , then $E (0) > 0$ .
Proof.
From Lemma 2.1, we can infer that
$\int_{Ω} F_{1} (u_{0} (x)) d x \leq \frac{1}{2} h_{1} (‖ \nabla u_{0} ‖_{L^{2} (Ω)}) ‖ \nabla u_{0} ‖_{L^{2} (Ω)}^{2} \leq \frac{1}{4} ‖ \nabla u_{0} ‖_{L^{2} (Ω)}^{2}$
and
$\int_{Ω} F_{2} (y_{0} (x)) d x \leq \frac{1}{2} h_{2} (‖ \nabla y_{0} ‖_{L^{2} (Ω)}) ‖ \nabla y_{0} ‖_{L^{2} (Ω)}^{2} \leq \frac{1}{4} ‖ \nabla y_{0} ‖_{L^{2} (Ω)}^{2} .$
As a consequence,
$\begin{aligned} E (0) & = \frac{1}{2} ‖ u_{1} ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} ‖ \nabla u_{0} ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} ‖ y_{1} ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} ‖ \nabla y_{0} ‖_{L^{2} (Ω)}^{2} \\ - \int_{Ω} F_{1} (u_{0} (x)) d x - \int_{Ω} F_{2} (y_{0} (x)) d x + \frac{1}{2} \int_{- τ}^{0} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s d x \\ > \frac{1}{4} ‖ u_{1} ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u_{0} ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{1} ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y_{0} ‖_{L^{2} (Ω)}^{2} \\ + \frac{1}{4} \int_{- τ}^{0} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s d x . \end{aligned}$

Therefore, since $U$ is a non-zero solution, we have proven that $E (0) > 0$ .

We can also obtain the following preliminary estimate.
Proposition 2.1
Let $(u, y)$ be a solution to (1.3) defined on an interval $[0, T),$ for some $T > 0.$ If $E (t) \geq \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2}$ , for any $t \in [0, T)$ , then
$E (t) \leq C (t) E (0),$
(2.8)
for any $t \geq 0$ , where
$C (t) = e^{4 ‖ a_{2} ‖_{L^{\infty} (Ω)} t} .$
(2.9)
Proof.
Differentiating formally $E$ with respect to $t$ , we obtain
$\begin{aligned} E^{'} (t) & = \int_{Ω} {u_{t} (x, t) u_{t t} (x, t) + \nabla u (x, t) \nabla u_{t} (x, t) + y_{t} (x, t) y_{t t} (x, t) + \nabla y (x, t) \nabla y_{t} (x, t)} d x \\ - \int_{Ω} u_{t} (x, t) f_{1} (u (x, t)) d x - \int_{Ω} y_{t} (x, t) f_{2} (y (x, t)) d x + \frac{1}{2} \int_{Ω} | a_{2} (x) | y_{t}^{2} (x, t) d x \\ - \frac{1}{2} \int_{Ω} | a_{2} (x) | y_{t}^{2} (x, t - τ) d x . \end{aligned}$

By using the equations satisfied by $u$ and $y$ in (1.3) and Green formula, we get
$\begin{aligned} E^{'} (t) & = \int_{Ω} {u_{t} (x, t) (Δ u (x, t) - b y_{t} (x, t) - a_{1} (x) u_{t} (x, t) + f_{1} (u (x, t))) + \nabla u (x, t) \nabla u_{t} (x, t) \\ + y_{t} (x, t) (Δ y (x, t) + b u_{t} (x, t) - a_{2} (x) y_{t} (x, t - τ) + f_{2} (y (x, t))) + \nabla y (x, t) \nabla y_{t} (x, t)} d x \\ - \int_{Ω} u_{t} (x, t) f_{1} (u (x, t)) d x - \int_{Ω} y_{t} (x, t) f_{2} (y (x, t)) d x + \frac{1}{2} \int_{Ω} | a_{2} (x) | y_{t}^{2} (x, t) d x \\ - \frac{1}{2} \int_{Ω} | a_{2} (x) | y_{t}^{2} (x, t - τ) d x \\ = - \int_{Ω} a_{1} (x) u_{t}^{2} (x, t) d x - \int_{Ω} a_{2} (x) y_{t} (x, t) y_{t} (x, t - τ) d x + \frac{1}{2} \int_{Ω} | a_{2} (x) | y_{t}^{2} (x, t) d x \\ - \frac{1}{2} \int_{Ω} | a_{2} (x) | y_{t}^{2} (x, t - τ) d x \\ \leq \int_{Ω} | a_{2} (x) | y_{t}^{2} (x, t) d x \leq ‖ a_{2} ‖_{L^{\infty} (Ω)} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2} . \end{aligned}$

Since we have assumed $E (t) \geq \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2}$ for any $t \in [0, T)$ , we obtain
$E^{'} (t) \leq 4 ‖ a_{2} ‖_{L^{\infty} (Ω)} E (t) .$
By Gronwall’s inequality, we then obtain (2.8).

Next, we can give the following estimate from below for sufficiently small initial data.
Lemma 2.4
Assume Hypothesis 2.1. Let $U$ be a non-trivial solution of (2.2) defined on an interval $[0, δ)$ , and fix a time $T \geq δ$ . Then, if $h_{1} (‖ \nabla u_{0} ‖_{L^{2} (Ω)}) < (1 / 2)$ , $h_{2} (‖ \nabla y_{0} ‖_{L^{2} (Ω)}) < (1 / 2)$ and $h_{i} (2 \sqrt{C (T) E (0)}) < (1 / 2)$ for $i = 1, 2$ , with $C (\cdot)$ defined as in (2.9), then
$\begin{aligned} E (t) & > \frac{1}{4} ‖ u_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y (t) ‖_{L^{2} (Ω)}^{2} \\ + \frac{1}{4} \int_{t - τ}^{t} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s \end{aligned}$
(2.10)
for all $t \in [0, δ)$ . In particular,
$E (t) > \frac{1}{4} ‖ U (t)) ‖_{H}^{2}, \forall t \in [0, δ) .$

Proof.
We argue by contradiction. Let us denote
$r := sup {s \in [0, δ) : (2.10) holds \forall t \in [0, s)} .$
We suppose by contradiction that $r < δ$ . Then, by continuity, we have
$\begin{aligned} E (r) & = \frac{1}{4} ‖ u_{t} (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{t} (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y (r) ‖_{L^{2} (Ω)}^{2} \\ + \frac{1}{4} \int_{r - τ}^{r} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s . \end{aligned}$

Then, we can see that
$E (r) \geq \frac{1}{4} ‖ y_{t} (r) ‖_{L^{2} (Ω)}^{2}, E (r) \geq \frac{1}{4} ‖ \nabla u (r) ‖_{L^{2} (Ω)}^{2} and E (r) \geq \frac{1}{4} ‖ \nabla y (r) ‖_{L^{2} (Ω)}^{2} .$
Using the fact that $h_{i}$ for $i = 1, 2$ are increasing functions and Proposition 2.1, we get
$h_{1} (‖ \nabla u (r) ‖_{L^{2} (Ω)}) \leq h_{1} (2 \sqrt{E (r)}) \leq h_{1} (2 \sqrt{C (r) E (0)}) < \frac{1}{2}$
and
$h_{2} (‖ \nabla y (r) ‖_{L^{2} (Ω)}) \leq h_{2} (2 \sqrt{E (r)}) \leq h_{2} (2 \sqrt{C (r) E (0)}) < \frac{1}{2} .$
Hence using the definition of the energy and (2.5), we conclude that
$\begin{aligned} E (r) & = \frac{1}{2} ‖ u_{t} (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} ‖ \nabla u (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} ‖ y_{t} (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} ‖ \nabla y (r) ‖_{L^{2} (Ω)}^{2} \\ - \int_{Ω} F_{1} (u (x, r)) d x - \int_{Ω} F_{2} (y (x, r)) d x + \frac{1}{2} \int_{r - τ}^{r} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s d x \\ > \frac{1}{4} ‖ u_{t} (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{t} (r) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y (r) ‖_{L^{2} (Ω)}^{2} \\ + \frac{1}{4} \int_{r - τ}^{r} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s d x . \end{aligned}$

This contradicts the maximality of $r$ . Hence, $r = δ$ and this concludes the proof of the lemma.

Next, we establish an exponential decay result for the abstract model (2.2) under an appropriate assumption on the function $a_{2}$ . Let us consider the following hypothesis
Hypothesis 2.2
The function $a_{2} \in L^{\infty} (Ω)$ satisfies
$e^{α τ} ‖ a_{2} ‖_{L^{\infty} (Ω)} < \frac{α}{M},$
with $M$ and $α$ as in (2.1).
Theorem 2.1
Assume that either Assumption 1.1 or Assumption 1.2 holds, together with Hypotheses 2.1 and 2.2. Let $U \in C ([0, T); H)$ a solution of the system (2.2) satisfying $‖ U (t) ‖_{H} \leq C_{ρ}$ for all $t \in [0, T)$ with $C_{ρ}$ such that $L (C_{ρ}) < α - M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ} / M$ . Then, the following exponential decay estimate holds
$‖ U (t) ‖_{H} \leq M ({‖ U_{0} ‖}_{H} + \int_{0}^{τ} e^{α s} ‖ Ψ (s - τ) ‖_{H} d s) e^{- (α - M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ} - M L (C_{ρ})) t},$
(2.11)
for any $t \in [0, T)$ .
Proof.
Applying Duhamel’s formula, we obtain
$\begin{aligned} ‖ U (t) ‖_{H} & \leq M e^{- α t} {‖ U_{0} ‖}_{H} + M e^{- α t} \int_{0}^{t} e^{α s} ‖ B U (s - τ) ‖_{H} d s \\ + M L (C_{ρ}) e^{- α t} \int_{0}^{t} e^{α s} ‖ U (s) ‖_{H} d s, \end{aligned}$
(2.12)
where we have used the assumption $‖ F (U (t)) ‖_{H} \leq L (C_{ρ}) ‖ U (t) ‖_{H}$ for all $t \geq 0$ .

We distinguish the following two cases:
Case $τ \leq t$ : Using (2.12), we obtain
$\begin{aligned} ‖ U (t) ‖_{H} & \leq M e^{- α t} {‖ U_{0} ‖}_{H} + M e^{- α t} \int_{0}^{τ} e^{α s} ‖ Ψ (s - τ) ‖_{H} d s \\ + M e^{- α t} \int_{τ}^{t} e^{α s} ‖ a_{2} ‖_{L^{\infty} (Ω)} ‖ U (s - τ) ‖_{H} d s + M L (C_{ρ}) e^{- α t} \int_{0}^{t} e^{α s} ‖ U (s) ‖_{H} d s . \end{aligned}$

By applying the change of variables $s - τ = z$ in the second integral, it follows that
$\begin{aligned} e^{α t} ‖ U (t) ‖_{H} & \leq M ‖ U_{0} ‖_{H} + M \int_{0}^{τ} e^{α s} ‖ Ψ (s - τ) ‖_{H} d s \\ + M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ} \int_{0}^{t} e^{α z} ‖ U (z) ‖_{H} d z + M L (C_{ρ}) \int_{0}^{t} e^{α s} ‖ U (s) ‖_{H} d s . \end{aligned}$

Defining $\tilde{u} (t) := e^{α t} ‖ U (t) ‖_{H}$ and $M_{0} := M ‖ U_{0} ‖_{H} + M \int_{0}^{τ} e^{α s} ‖ Ψ (s - τ) ‖_{H} d s$ , we get
$\tilde{u} (t) \leq M_{0} + \int_{0}^{t} (M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ} + M L (C_{ρ})) \tilde{u} (s) d s .$
By Gronwall’s inequality, it follows that
$\tilde{u} (t) \leq M_{0} e^{(M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ} + M L (C_{ρ})) t} .$
Using the definition of $\tilde{u}$ and Assumption 2.2, we obtain (2.11).

Case $0 \leq t < τ$ : From (2.12), it follows that
$\begin{aligned} ‖ U (t) ‖_{H} & \leq M e^{- α t} {‖ U_{0} ‖}_{H} + M e^{- α t} \int_{0}^{τ} e^{α s} ‖ Ψ (s - τ) ‖_{H} d s \\ + M L (C_{ρ}) e^{- α t} \int_{0}^{t} e^{α s} ‖ U (s) ‖_{H} d s . \end{aligned}$

Thus,
$\tilde{u} (t) \leq M_{0} + \int_{0}^{t} M L (C_{ρ}) \tilde{u} (s) d s .$
By Gronwall’s inequality, we obtain
$\tilde{u} (t) \leq M_{0} e^{M L (C_{ρ}) t} .$
As before, using Assumption 2.2, we obtain (2.11).

We are now ready to prove the global existence and uniqueness for solutions of system (1.3).
Theorem 2.2
Assume that either Assumption 1.1 or Assumption 1.2 holds, together with Hypotheses 2.1 and 2.2. We consider the initial data $U_{0} \in H$ and $Ψ \in C ([- τ, 0]; H)$ such that
${‖ U_{0} ‖}_{H}^{2} + \int_{0}^{τ} ‖ \sqrt{| a_{2} |} g (s - τ) ‖_{L^{2} (Ω)}^{2} d s < ρ^{2},$
(2.13)
with $ρ$ sufficiently small constant. Then, the problem (2.2) has a unique global solution exponentially decaying according to (2.11).
Proof.
Let us fix a time $T > τ$ such that
$C_{T} := 2 M^{2} (1 + τ e^{2 α τ} ‖ a_{2} ‖_{\infty}) (1 + τ ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ}) e^{- (α - M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ}) T} < 1.$
Moreover let $ρ > 0$ be such that
$ρ \leq \frac{1}{2 \sqrt{C (T)}} min_{i = 1, 2} {h_{i}^{- 1} (\frac{1}{2})} .$
We consider initial data such that
$‖ u_{1} ‖_{L^{2} (Ω)}^{2} + ‖ \nabla u_{0} ‖_{L^{2} (Ω)}^{2} + ‖ y_{1} ‖_{L^{2} (Ω)}^{2} + ‖ \nabla y_{0} ‖_{L^{2} (Ω)}^{2} + \int_{- τ}^{0} ‖ \sqrt{| a_{2} |} g (s) ‖_{L^{2} (Ω)}^{2} d s < ρ^{2} .$
We observe that this is equivalent to
$‖ U_{0} ‖_{H}^{2} + \int_{0}^{τ} ‖ \sqrt{| a_{2} |} g (s - τ) ‖_{L^{2} (Ω)}^{2} d s < ρ^{2} .$
(2.14)
Now, by Lemma 2.2 we know that there exists a local solution $(u, y)$ of (1.3) on a time interval $[0, δ)$ . Without loss of generality, we can assume $δ < T$ (eventually, we can take a larger $T$ ). From our assumption on the initial data, on the monotonicity of $h_{i}$ for $i = 1, 2$ and the fact that $C (T) > 1$ , we have
$h_{1} (‖ \nabla u_{0} ‖_{L^{2} (Ω)}) \leq h_{1} (ρ) \leq h_{1} (\frac{1}{2 \sqrt{C (T)}} h_{1}^{- 1} (\frac{1}{2})) < \frac{1}{2}$
and
$h_{2} (‖ \nabla y_{0} ‖_{L^{2} (Ω)}) \leq h_{2} (ρ) \leq h_{2} (\frac{1}{2 \sqrt{C (T)}} h_{2}^{- 1} (\frac{1}{2})) < \frac{1}{2} .$
Hence, by Lemma 2.3, $E (0) > 0$ . Moreover, from (2.5), we obtain
$\begin{aligned} E (0) & \leq \frac{1}{2} ‖ u_{1} ‖_{L^{2} (Ω)}^{2} + \frac{3}{4} ‖ \nabla u_{0} ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} ‖ y_{1} ‖_{L^{2} (Ω)}^{2} \\ + \frac{3}{4} ‖ \nabla y_{0} ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} \int_{- τ}^{0} ‖ \sqrt{| a_{2} |} g (s) ‖_{L^{2} (Ω)}^{2} d s d x \leq ρ^{2}, \end{aligned}$
(2.15)
which gives us
$h_{i} (2 \sqrt{C (T) E (0)}) < h_{i} (2 \sqrt{C (T)} ρ) < h_{i} (h_{i}^{- 1} (\frac{1}{2})) = \frac{1}{2}$
for $i = 1, 2$ . Hence, we can apply Lemma 2.4 to obtain
$\begin{aligned} E (t) & > \frac{1}{4} ‖ u_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y (t) ‖_{L^{2} (Ω)}^{2} \\ + \frac{1}{4} \int_{t - τ}^{t} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s > 0 \end{aligned}$
(2.16)
for all $t \in [0, δ)$ ; in particular $E (t) \geq \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2}$ for all $t \in [0, δ)$ . Thus, applying Proposition 2.1 and the fact that $δ < T$ we have
$E (t) \leq C (t) E (0), \forall t \in [0, δ) .$
(2.17)
As a consequence, from (2.16) and (2.17), we obtain
$\begin{aligned} \frac{1}{4} ‖ \nabla u (t) ‖_{L^{2} (Ω)}^{2} & \leq \frac{1}{4} ‖ u_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y (t) ‖_{L^{2} (Ω)}^{2} \\ + \frac{1}{4} \int_{t - τ}^{t} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s < E (t) \leq C (T) E (0), \end{aligned}$
and
$\begin{aligned} \frac{1}{4} ‖ \nabla y (t) ‖_{L^{2} (Ω)}^{2} & \leq \frac{1}{4} ‖ u_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y (t) ‖_{L^{2} (Ω)}^{2} \\ + \frac{1}{4} \int_{t - τ}^{t} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s < E (t) \leq C (T) E (0), \end{aligned}$
for all $t \in [0, δ)$ . Then, we can extend the solution in $t = δ$ and on the whole interval $[0, T]$ . In particular, for $t = T$ , we get
$h_{1} (‖ \nabla u (T) ‖_{L^{2} (Ω)}) \leq h_{1} (2 \sqrt{C (T) E (0)}) < \frac{1}{2}$
and
$h_{2} (‖ \nabla y (T) ‖_{L^{2} (Ω)}) \leq h_{2} (2 \sqrt{C (T) E (0)}) < \frac{1}{2} .$
By (2.16) and (2.15) we deduce
$\frac{1}{4} ‖ U (t) ‖_{H}^{2} \leq E (t) \leq C (T) E (0) \leq C (T) ρ^{2}, \forall t \in [0, T],$
i.e
$‖ U (t) ‖_{H} \leq C_{ρ}, \forall t \in [0, T],$
where $C_{ρ} := 2 \sqrt{C (T)} ρ$ . Now, eventually choosing smaller values of $ρ$ , we suppose that $ρ$ is such that $L (C_{ρ}) < (α - M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ}) / 2 M$ . Hence, Theorem 2.1 gives us the following estimate:
$‖ U (t) ‖_{H} \leq M ({‖ U_{0} ‖}_{H} + \int_{0}^{τ} e^{α s} ‖ Ψ (s - τ) ‖_{H} d s) e^{- \frac{(α - M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ})}{2} t},$
(2.18)
for any $t \in [0, T]$ . By Hypothesis (2.14) and the Hölder inequality, it follows
$\int_{0}^{τ} e^{α s} ‖ Ψ (s - τ) ‖_{H} d s = \int_{0}^{τ} e^{α s} ‖ a_{2} g (s - τ) ‖_{L^{2} (Ω)} d s \leq e^{α τ} \sqrt{τ} ρ ‖ a_{2} ‖_{\infty}^{1 / 2} .$
Hence, from (2.18), we obtain
$‖ U (t) ‖_{H}^{2} \leq 2 M^{2} ρ^{2} (1 + τ e^{2 α τ} ‖ a_{2} ‖_{\infty}) e^{- (α - M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ}) t} .$
for any $t \in [0, T]$ . Moreover
$\int_{T - τ}^{T} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s \leq 2 M^{2} τ ρ^{2} ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ} (1 + τ e^{2 α τ} ‖ a_{2} ‖_{\infty}) e^{- (α - M ‖ a_{2} ‖_{L^{\infty} (Ω)} e^{α τ}) T} .$
Then,
$‖ U (T) ‖_{H}^{2} + \int_{T - τ}^{T} ‖ \sqrt{| a_{2} |} y_{t} (s) ‖_{L^{2} (Ω)}^{2} d s \leq C_{T} ρ^{2} < ρ^{2},$
where we have used the fact that $C_{T} < 1$ .

We proceed by applying a similar argument as before on the interval $[T, 2 T]$ , thereby extending the solution to $[0, 2 T]$ . By iterating this process inductively, we construct a unique global solution to (2.2). Moreover, at each step, the exponential decay estimate (2.11) is preserved, ensuring that the global solution satisfies this decay property.
3. The Coupled System in the Presence of Indefinite Damping

In this section, we study the problem (1.4). Referring to Section 2, one can reproduce the same procedure in the case when $τ = 0$ , and deduce that the problem (1.4) is exponentially stable under the following condition on the coefficient $a_{2}$ :

‖ a_{2} ‖_{L^{\infty} (Ω)} < \frac{α}{M} .

The main objective of this section is to establish the exponential stability of the problem (1.4) under a weaker condition, namely, by restricting the assumption on the negative part of

a_{2}

, that is

a_{2}^{-} := - min {a_{2}, 0}

Before addressing problem (1.4), we first need to investigate the exponential stability of the associated coupled wave equation with a definite (i.e., non-negative) damping:

{\begin{cases} u_{t t} (x, t) - Δ u (x, t) + b (x) y_{t} (x, t) + d_{1} (x) u_{t} (x, t) = 0, & in Ω \times (0, \infty), \\ y_{t t} (x, t) - Δ y (x, t) - b (x) u_{t} (x, t) + d_{2} (x) y_{t} (x, t) = 0, & in Ω \times (0, \infty), \\ u (x, t) = y (x, t) = 0, & on \partial Ω \times (0, \infty), \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), & in Ω, \\ y (x, 0) = y_{0} (x), y_{t} (x, 0) = y_{1} (x), & in Ω, \end{cases}

(3.1)

where

d_{1} \in C^{0} (\bar{Ω}, R)

and

d_{2} \in L^{\infty} (Ω)

are given functions. The coefficient

d_{1}

satisfies the conditions

d_{1} (x) \geq 0 a.e in Ω,

and

d_{1} (x) > d_{0} > 0 a.e in a subdomain ω \subset Ω .

While the coefficient

d_{2}

satisfies the condition

d_{2} (x) \geq 0 a.e in Ω .

Next, we define the linear unbounded operator

\hat{A} : D (\hat{A}) \subset H \to H

{\begin{cases} D (\hat{A}) = {((H^{2} (Ω) \cap H_{0}^{1} (Ω)) \times H_{0}^{1} (Ω))}^{2}, \\ \hat{A} U = (u_{2}, Δ u_{1} - b y_{2} - d_{1} u_{2}, y_{2}, Δ y_{1} + b u_{2} - d_{2} y_{2}), \forall U = (u_{1}, u_{2}, y_{1}, y_{2}) \in D (\hat{A}) . \end{cases}

(3.2)

Using this operator, we rewrite the system (3.1) as an abstract Cauchy problem. Indeed, setting

U (t) = (u (t), u_{t} (t), y (t), y_{t} (t))

, one has that (3.1) can be formulated as

{\begin{cases} U^{'} (t) = \hat{A} U (t), t > 0, \\ U (0) = U_{0}, \end{cases}

(3.3)

where

U (0) = (u_{0}, u_{1}, y_{0}, y_{1})

The existence and uniqueness result reads as follows.

Theorem 3.1

For any $U_{0} = (u_{0}, u_{1}, y_{0}, y_{1}) \in H$ , there exists a unique solution $U = (u, u_{t}, y, y_{t}) \in C^{0} ([0, \infty); H)$ of system (3.3). Moreover, if $U_{0} = (u_{0}, u_{1}, y_{0}, y_{1}) \in D (\hat{A})$ , then

U \in C^{0} ([0, \infty); D (\hat{A})) \cap C^{1} ([0, \infty); H) .

Proof.

We will prove that the operator $\hat{A}$ defined in (3.2) generates a contraction semi-group on the Hilbert space $H$ . To this end, we show that the unbounded operator $\hat{A}$ is maximal dissipative on $H$ . According to Cazenave and Haraux (1998, Proposition 2.2.6), it is sufficient to prove that $\hat{A}$ is dissipative and that $I - \hat{A}$ is surjective, where

I = diag (I d, I d, I d, I d) .

$\underline{\hat{A} is dissipative}$ . Let $U = (u_{1}, u_{2}, y_{1}, y_{2}) \in D (\hat{A})$ . Then, by integrating by parts and due to the fact that $d_{1} (x) \geq 0$ and $d_{2} (x) \geq 0$ , we obtain

\begin{aligned} ⟨ \hat{A} U, U ⟩_{H} & = ⟨ (u_{2}, Δ u_{1} - b y_{2} - d_{1} u_{2}, y_{2}, Δ y_{1} + b u_{2} - d_{2} y_{2}), (u_{1}, u_{2}, y_{1}, y_{2}) ⟩_{H} \\ = \int_{Ω} (\nabla u_{1} \nabla u_{2} + (Δ u_{1} - b y_{2} - d_{1} u_{2}) u_{2} + \nabla y_{1} \nabla y_{2} + (Δ y_{1} + b u_{2} - d_{2} y_{2}) y_{2}) d x \\ = - \int_{Ω} d_{1} (x) u_{2}^{2} (x) d x - \int_{Ω} d_{2} (x) y_{2}^{2} (x) d x \\ \leq 0. \end{aligned}

Therefore the operator

\hat{A}

is dissipative.

$I - \hat{A}$ is surjective. Given $F = (f_{1}, f_{2}, f_{3}, f_{4}) \in H$ , we seek $U = (u_{1}, u_{2}, y_{1}, y_{2}) \in D (\hat{A})$ such that

U - \hat{A} U = F \Leftrightarrow {\begin{cases} u_{1} - u_{2} & = f_{1}, \\ u_{2} - Δ u_{1} + b y_{2} + d_{1} u_{2} & = f_{2}, \\ y_{1} - y_{2} & = f_{3}, \\ y_{2} - Δ y_{1} - b u_{2} + d_{2} y_{2} & = f_{4} . \end{cases}

(3.4)

We suppose that we have found

u_{1}

and

y_{1}

with the right regularity. Then, we set

u_{2} = u_{1} - f_{1} and y_{2} = y_{1} - f_{3} .

(3.5)

Eliminating

u_{2}

and

y_{2}

in (3.4), we get the following system

{\begin{cases} u_{1} - Δ u_{1} + b y_{1} + d_{1} u_{1} = (1 + d_{1}) f_{1} + f_{2} + b f_{3}, \\ y_{1} - Δ y_{1} - b u_{1} + d_{2} y_{1} = (1 + d_{2}) f_{3} - b f_{1} + f_{4} . \end{cases}

(3.6)

Now, we consider the bilinear form

Γ : (H_{0}^{1} (Ω) \times H_{0}^{1} (Ω))^{2} \to R

given by

\begin{aligned} Γ ((u_{1}, y_{1}), (ϕ_{1}, ϕ_{2})) & = \int_{Ω} (u_{1} ϕ_{1} + \nabla u_{1} \nabla ϕ_{1} + y_{1} ϕ_{2} + \nabla y_{1} \nabla ϕ_{2}) d x + \int_{Ω} d_{1} u_{1} ϕ_{1} d x \\ + \int_{Ω} d_{2} y_{1} ϕ_{2} d x + \int_{Ω} b (y_{1} ϕ_{1} - u_{1} ϕ_{2}) d x \end{aligned}

and the linear form

L : H_{0}^{1} (Ω) \times H_{0}^{1} (Ω) \to R

given by

L (ϕ_{1}, ϕ_{2}) = \int_{Ω} ((1 + d_{1}) f_{1} + f_{2} + b f_{3}) ϕ_{1} d x + \int_{Ω} ((1 + d_{2}) f_{3} - b f_{1} + f_{4}) ϕ_{2} d x .

In view of Poincaré inequality,

Γ

is a continuous bilinear form on

H_{0}^{1} (Ω) \times H_{0}^{1} (Ω)

and

L

is a continuous linear functional on

H_{0}^{1} (Ω) \times H_{0}^{1} (Ω)

. Moreover, it is easy to check that

Γ

is also coercive on

H_{0}^{1} (Ω) \times H_{0}^{1} (Ω)

. Indeed, let

(u_{1}, y_{1}) \in H_{0}^{1} (Ω) \times H_{0}^{1} (Ω)

\begin{aligned} Γ ((u_{1}, y_{1}), (u_{1}, y_{1})) & = \int_{Ω} (u_{1}^{2} + | \nabla u_{1} |^{2} + y_{1}^{2} + | \nabla y_{1} |^{2}) d x + \int_{Ω} d_{1} u_{1}^{2} d x + \int_{Ω} d_{2} y_{1}^{2} d x \\ \geq ‖ (u_{1}, y_{1}) ‖_{H_{0}^{1} (Ω) \times H_{0}^{1} (Ω)}^{2} . \end{aligned}

By applying the Lax–Milgram theorem, we deduce that there exists a unique solution

(u_{1}, y_{1}) \in H_{0}^{1} (Ω) \times H_{0}^{1} (Ω)

Γ ((u_{1}, y_{1}), (ϕ_{1}, ϕ_{2})) = L (ϕ_{1}, ϕ_{2}) for all (ϕ_{1}, ϕ_{2}) \in H_{0}^{1} (Ω) \times H_{0}^{1} (Ω) .

(3.7)

(ϕ_{1}, ϕ_{2}) \in D (Ω) \times D (Ω)

, then

(u_{1}, y_{1})

solves (3.6) in

D^{'} (Ω) \times D^{'} (Ω)

and so

(u_{1}, y_{1}) \in (H^{2} (Ω))^{2}

(See Allaire, 2009, Chapter 5). Now, take

(u_{2}, y_{2}) := (u_{1} - f_{1}, y_{1} - f_{3})

; then

(u_{2}, y_{2}) \in (H_{0}^{1} (Ω))^{2}

. Thus, we have found

U = (u_{1}, u_{2}, y_{1}, y_{2}) \in D (\hat{A})

, which satisfies (3.4). Consequently

I - \hat{A}

is surjective. Conclusively, the Lumer–Phillips Theorem implies the operator

\hat{A}

generates a strongly continuous semigroup of contraction in

H

. Consequently, the well-posedness result follows from the Hille-Yosida theorem (see Barbu, 1998, Theorem 4.5.1 or Coron, 2007, Theorem A.11).

Now, we study the exponential stability of system (3.1). To this aim we first define its corresponding energy by

E (t) := \frac{1}{2} \int_{Ω} {u_{t}^{2} (x, t) + | \nabla u (x, t) |^{2} + y_{t}^{2} (x, t) + | \nabla y (x, t) |^{2}} d x, \forall t \geq 0.

(3.8)

Direct computations allow to show that the energy exhibits a decay with respect to time. More precisely, we have the following result. The energy $E$ associated to the solution of system (3.1) satisfies

E^{'} (t) = - \int_{Ω} d_{1} (x) u_{t}^{2} (x, t) d x - \int_{Ω} d_{2} (x) y_{t}^{2} (x, t) d x, \forall t \geq 0.

(3.9)

Consequently, the energy

E (.)

is nonincreasing.

We can prove the following exponential stability result for the problem (3.1).

Theorem 3.2

Assume that either Assumption 1.1 or Assumption 1.2 holds. There exist two positive constants $\hat{M}, \hat{α}$ such that

E (t) \leq \hat{M} e^{- \hat{α} t} E (0), t > 0,

(3.10)

for any solution to the problem (3.1).

Proof.

As in Nicaise and Pignotti (2006), by following (Zuazua, 1990), we can decompose the solution $(u, y)$ of (3.1) as $u = w + \tilde{w}$ and $y = θ + \tilde{θ},$ where $(w, θ)$ is the solution to the problem

{\begin{cases} w_{t t} (x, t) - Δ w (x, t) + b θ_{t} (x, t) + d_{1} w_{t} (x, t) = 0, & in Ω \times (0, \infty), \\ θ_{t t} (x, t) - Δ θ (x, t) - b w_{t} (x, t) = 0, & in Ω \times (0, \infty), \\ w (x, t) = θ (x, t) = 0, & on \partial Ω \times (0, \infty), \\ w (x, 0) = u_{0} (x), w_{t} (x, 0) = u_{1} (x), & in Ω, \\ θ (x, 0) = y_{0} (x), θ_{t} (x, 0) = y_{1} (x), & in Ω, \end{cases}

(3.11)

and

(\tilde{w}, \tilde{θ})

solution to the problem

{\begin{cases} {\tilde{w}}_{t t} (x, t) - Δ \tilde{w} (x, t) + b {\tilde{θ}}_{t} (x, t) + d_{1} {\tilde{w}}_{t} (x, t) = 0, & in Ω \times (0, \infty), \\ {\tilde{θ}}_{t t} (x, t) - Δ \tilde{θ} (x, t) - b {\tilde{w}}_{t} (x, t) = - d_{2} (x) y_{t} (x, t), & in Ω \times (0, \infty), \\ \tilde{w} (x, t) = \tilde{θ} (x, t) = 0, & on \partial Ω \times (0, \infty), \\ \tilde{w} (x, 0) = 0, {\tilde{w}}_{t} (x, 0) = 0, & in Ω, \\ \tilde{θ} (x, 0) = 0, {\tilde{θ}}_{t} (x, 0) = 0, & in Ω . \end{cases}

(3.12)

Thus, as a direct consequence of Haraux (1989, Proposition 2), it follows that under either Assumption 1.1 or Assumption 1.2, the exponential stability property given by (2.1) implies the existence of a constant time

T_{0} > 0

such that, for every

T > T_{0}

, the following holds:

E (0) \leq C \int_{0}^{T} \int_{Ω} d_{1} (x) w_{t}^{2} (x, t) d x d t \leq 2 C \int_{Ω} d_{1} (x) \int_{0}^{T} (u_{t}^{2} (x, t) + {\tilde{w}}_{t}^{2} (x, t)) d t d x .

(3.13)

Now, observe that from (3.12), it follows that

\begin{aligned} \frac{1}{2} \frac{d}{d t} & \int_{Ω} {{\tilde{w}}_{t}^{2} (x, t) + | \nabla \tilde{w} (x, t) |^{2} + {\tilde{θ}}_{t}^{2} (x, t) + | \nabla \tilde{θ} (x, t) |^{2}} d x \\ = \int_{Ω} {{\tilde{w}}_{t} (x, t) {\tilde{w}}_{t t} (x, t) + \nabla \tilde{w} (x, t) \nabla {\tilde{w}}_{t} (x, t) + {\tilde{θ}}_{t} (x, t) {\tilde{θ}}_{t t} (x, t) + \nabla \tilde{θ} (x, t) \nabla {\tilde{θ}}_{t} (x, t)} d x \\ = \int_{Ω} (- d_{1} (x) {\tilde{w}}_{t}^{2} (x, t) - d_{2} (x) {\tilde{θ}}_{t} (x, t) y_{t} (x, t)) d x . \end{aligned}

Integrating with respect to time on

[0, t]

, for

t \in (0, T]

, and using the fact that the initial data in (3.12) is set to zero, we have

\begin{array}{l} \frac{1}{2} \int_{Ω} {{\tilde{w}}_{t}^{2} (x, t) + | \nabla \tilde{w} (x, t) |^{2} + {\tilde{θ}}_{t}^{2} (x, t) + | \nabla \tilde{θ} (x, t) |^{2}} d x \\ = - \int_{Ω} d_{1} (x) \int_{0}^{t} {\tilde{w}}_{t}^{2} (x, t) d t d x - \int_{Ω} d_{2} (x) \int_{0}^{t} {\tilde{θ}}_{t} (x, t) y_{t} (x, t) d t d x \\ \leq \int_{Ω} d_{2} (x) \int_{0}^{T} | {\tilde{θ}}_{t} (x, t) y_{t} (x, t) | d t d x . \end{array}

By integrating with respect to time, we deduce that

\begin{array}{l} \int_{0}^{T} \int_{Ω} {{\tilde{w}}_{t}^{2} (x, t) + {\tilde{θ}}_{t}^{2} (x, t)} d x d t \\ \leq \frac{1}{2} \int_{Ω} \int_{0}^{T} {\tilde{θ}}_{t}^{2} (x, t) d t d x + 2 T^{2} ‖ d_{2} ‖_{L^{\infty} (Ω)} \int_{Ω} d_{2} (x) \int_{0}^{T} y_{t}^{2} (x, t) d t d x, \end{array}

which implies that

\int_{0}^{T} \int_{Ω} {\tilde{w}}_{t}^{2} (x, t) d x d t \leq 2 T^{2} ‖ d_{2} ‖_{L^{\infty} (Ω)} \int_{Ω} d_{2} (x) \int_{0}^{T} y_{t}^{2} (x, t) d t d x .

This combined with (3.13) yield that

E (0) \leq C_{0} (\int_{Ω} d_{1} (x) \int_{0}^{T} u_{t}^{2} (x, t) d x d t + \int_{Ω} d_{2} (x) \int_{0}^{T} y_{t}^{2} (x, t) d x d t),

where

C_{0}

is a positive constant.

From (3.9), we have

E (T) - E (0) = - (\int_{Ω} d_{1} (x) \int_{0}^{T} u_{t}^{2} (x, t) d t d x + \int_{Ω} d_{2} (x) \int_{0}^{T} y_{t}^{2} (x, t) d t d x) .

Hence, we obtain

E (T) \leq E (0) \leq C_{0} (\int_{Ω} d_{1} (x) \int_{0}^{T} u_{t}^{2} (x, t) d t d x + \int_{Ω} d_{2} (x) \int_{0}^{T} y_{t}^{2} (x, t) d t d x) = C_{0} (E (0) - E (T)),

then

E (T) \leq \hat{C} E (0)

with

\hat{C} < 1

This easily implies the stability estimate (3.10), since our system (3.1) is invariant by translation and the energy $E$ is decreasing.

Returning to the problem (1.4), and using the same notations as those introduced in Section 2, we define the linear operator $\tilde{A} : D (\tilde{A}) \subset H \to H$ by

D (\tilde{A}) = {((H^{2} (Ω) \cap H_{0}^{1} (Ω)) \times H_{0}^{1} (Ω))}^{2}, \tilde{A} U = [\begin{matrix} u_{2} \\ Δ u_{1} - b y_{2} - a_{1} u_{2} \\ y_{2} \\ Δ y_{1} + b u_{2} - a_{2}^{+} y_{2} \end{matrix}],

where

a_{2}^{+}

denotes the positive part of the function

a_{2}

, defined by

a_{2}^{+} (x) := max {a_{2} (x), 0}, \forall x \in Ω .

By applying Theorem 3.2 with $d_{1} = a_{1}$ and $d_{2} = a_{2}^{+}$ , we deduce that the operator $\tilde{A}$ is non-negative and generates a contraction semigroup $(\tilde{S} (t))_{t \geq 0}$ on $H$ , which is exponentially stable under either Assumption 1.1 or Assumption 1.2. More precisely, there exist constants $\tilde{M} > 0$ and $\tilde{α} > 0$ such that

‖ \tilde{S} (t) ‖_{L (H)} \leq \tilde{M} e^{- \tilde{α} t}, \forall t \geq 0.

Consequently, the coupled system (1.4) can be reformulated as the following abstract Cauchy problem:

{\begin{cases} \dot{U} (t) = \tilde{A} U (t) + a_{2}^{-} P U (t) + F (U (t)), & t > 0, \\ U (0) = U_{0}, \end{cases}

(3.14)

where

P \in R^{4 \times 4}

is the linear projection operator defined by

P := (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .

Observe that, if $U_{0} \in H$ , then the following Duhamel formula holds:

U (t) = \tilde{S} (t) U_{0} + \int_{0}^{t} \tilde{S} (t - s) a_{2}^{-} P U (s) d s + \int_{0}^{t} \tilde{S} (t - s) F (U (s)) d s .

(3.15)

Using Cazenave and Haraux (1998, Proposition 4.3.3), we can establishing the local existence and uniqueness of solutions, as stated in the following lemma.

Lemma 3.1

Let $U_{0} \in H$ . Then, the system (3.14) has a unique continuous local solution $U$ defined on a time interval $[0, δ)$ .

Let us introduce the following energy functional associated to system (3.14):

Definition 3.1

Let $(u, y)$ be a mild solution of (1.4). Let us define its energy by

\begin{aligned} \tilde{E} (t) & = \frac{1}{2} \int_{Ω} {u_{t}^{2} (x, t) + | \nabla u (x, t) |^{2} + y_{t}^{2} (x, t) + | \nabla y (x, t) |^{2}} d x \\ - \int_{Ω} F_{1} (u (x, t)) d x - \int_{Ω} F_{2} (y (x, t)) d x . \end{aligned}

(3.16)

By arguing as in the proof of Lemma 2.3, we can establish the following result concerning the indefinite damping case.

Lemma 3.2

Assume that Hypothesis 2.1 holds. Let $U$ be a non-trivial solution of the problem (3.14), defined on the interval $[0, T) .$ If the initial data satisfy

h_{1} (‖ \nabla u_{0} ‖_{L^{2} (Ω)}) < \frac{1}{2} and h_{2} (‖ \nabla y_{0} ‖_{L^{2} (Ω)}) < \frac{1}{2},

then the initial energy is strictly positive, that is,

\tilde{E} (0) > 0

We also have the following result.

Proposition 3.1

Let $(u, y)$ be a solution to (1.4). If $\tilde{E} (t) \geq (1 / 4) ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2}$ , for any $t \geq 0$ , then

\tilde{E} (t) \leq \tilde{C} (t) \tilde{E} (0),

(3.17)

for any

t \geq 0

, where

\tilde{C} (t) = e^{4 ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)} t} .

(3.18)

Proof.

Differentiating formally $\tilde{E}$ with respect to $t$ , we obtain

\begin{aligned} {\tilde{E}}^{'} (t) & = \int_{Ω} {u_{t} (x, t) u_{t t} (x, t) + \nabla u (x, t) \nabla u_{t} (x, t) + y_{t} (x, t) y_{t t} (x, t) + \nabla y (x, t) \nabla y_{t} (x, t)} d x \\ - \int_{Ω} f_{1} (u (x, t)) u_{t} (x, t) d x - \int_{Ω} f_{2} (y (x, t)) y_{t} (x, t) d x . \end{aligned}

By using the equations satisfied by $u$ and $y$ in (1.3) and Green formula, we get

\begin{aligned} {\tilde{E}}^{'} (t) & = \int_{Ω} {u_{t} (x, t) (Δ u (x, t) - b y_{t} (x, t) - a_{1} (x) u_{t} (x, t) + f_{1} (u (x, t))) + \nabla u (x, t) \nabla u_{t} (x, t) \\ + y_{t} (x, t) (Δ y (x, t) + b u_{t} (x, t) - a_{2} (x) y_{t} (x, t) + f_{2} (y (x, t))) + \nabla y (x, t) \nabla y_{t} (x, t)} d x \\ - \int_{Ω} u_{t} (x, t) f_{1} (u (x, t)) d x - \int_{Ω} y_{t} (x, t) f_{2} (y (x, t)) d x \\ = - \int_{Ω} a_{1} (x) u_{t}^{2} (x, t) d x - \int_{Ω} a_{2} (x) y_{t}^{2} (x, t) d x \\ = - \int_{Ω} a_{1} (x) u_{t}^{2} (x, t) d x - \int_{Ω} a_{2}^{+} (x) y_{t}^{2} (x, t) d x + \int_{Ω} a_{2}^{-} (x) y_{t}^{2} (x, t) d x \\ \leq \int_{Ω} a_{2}^{-} (x) y_{t}^{2} (x, t) d x \leq ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2} . \end{aligned}

Since we have assumed $\tilde{E} (t) \geq (1 / 4) ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2}$ for any $t \geq 0$ , we obtain

{\tilde{E}}^{'} (t) \leq 4 ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)} \tilde{E} (t) .

By Gronwall inequality we then obtain (3.17).

Using Proposition 3.1 and adapting the arguments used in the proof of Lemma 2.4, we obtain the following result concerning the lower bound of the energy in the presence of indefinite damping.

Lemma 3.3

Assume that Hypothesis 2.1 holds. Let $U$ be a non-trivial solution of problem (3.14), defined on the interval $[0, δ)$ and fix a time $T \geq δ$ .

Suppose the initial data $(u_{0}, u_{1}, y_{0}, y_{1})$ satisfy the following conditions:

h_{1} (‖ \nabla u_{0} ‖_{L^{2} (Ω)}) < \frac{1}{2}, h_{2} (‖ \nabla y_{0} ‖_{L^{2} (Ω)}) < \frac{1}{2}, and h_{i} (2 \sqrt{\tilde{C} (T) \tilde{E} (0)}) < \frac{1}{2} for i = 1, 2,

where

\tilde{C} (\cdot)

is the function defined in (3.18).

Then, the energy $\tilde{E} (t)$ satisfies the following strict lower bound for all $t \in [0, δ)$ :

\tilde{E} (t) > \frac{1}{4} ‖ u_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y (t) ‖_{L^{2} (Ω)}^{2} .

(3.19)

In particular, one has:

\tilde{E} (t) > \frac{1}{4} ‖ U (t) ‖_{H}^{2}, \forall t \in [0, δ) .

Next, we establish an exponential decay result for the abstract system (3.14) under a weaker condition on the damping coefficient $a_{2}$ . More precisely, we impose a smallness assumption only on the negative part of $a_{2}$ . To this end, we introduce the following hypothesis:

Hypothesis 3.1

The function $a_{2} \in L^{\infty} (Ω)$ satisfies

‖ a_{2}^{-} ‖_{L^{\infty} (Ω)} < \frac{\tilde{α}}{\tilde{M}} .

Under this assumption, we prove that the solution to (3.14) decays exponentially in time. This is stated precisely in the following theorem.

Theorem 3.3

Assume that either Assumption 1.1 or Assumption 1.2 holds, together with Hypotheses 2.1 and 3.1. Let $U \in C ([0, T); H)$ be a solution of the problem (3.14) satisfying the uniform bound

‖ U (t) ‖_{H} \leq C_{ρ}, \forall t \in [0, T),

for some constant

C_{ρ} > 0

such that

L (C_{ρ}) < \frac{\tilde{α} - \tilde{M} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)}}{\tilde{M}},

where

L (\cdot)

is the Lipschitz constant of the nonlinearity as defined in Hypothesis 2.1.

Then, the following exponential decay estimate holds:

‖ U (t) ‖_{H} \leq \tilde{M} ‖ U_{0} ‖_{H} e^{- (\tilde{α} - \tilde{M} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)} - \tilde{M} L (C_{ρ})) t}, \forall t \in [0, T) .

(3.20)

Proof.

Applying Duhamel’s formula (3.15), we obtain

\begin{aligned} ‖ U (t) ‖_{H} & \leq \tilde{M} e^{- \tilde{α} t} {‖ U_{0} ‖}_{H} + \tilde{M} e^{- \tilde{α} t} \int_{0}^{t} e^{\tilde{α} s} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)} ‖ P U (s) ‖_{H} d s \\ + \tilde{M} e^{- \tilde{α} t} L (C_{ρ}) \int_{0}^{t} e^{\tilde{α} s} ‖ U (s) ‖_{H} d s, \end{aligned}

(3.21)

where we have used the assumption

‖ F (U (t)) ‖_{H} \leq L (C_{ρ}) ‖ U (t) ‖_{H}

for all

t \geq 0

It follows that

\begin{aligned} e^{\tilde{α} t} ‖ U (t) ‖_{H} \leq \tilde{M} {‖ U_{0} ‖}_{H} + (\tilde{M} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)} + \tilde{M} L (C_{ρ})) \int_{0}^{t} e^{\tilde{α} s} ‖ U (s) ‖_{H} d s . \end{aligned}

By Gronwall’s inequality, we obtain

e^{\tilde{α} t} ‖ U (t) ‖_{H} \leq \tilde{M} {‖ U_{0} ‖}_{H} e^{(\tilde{M} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)} + \tilde{M} L (C_{ρ})) t} .

As before, using Assumption 3.1, we obtain (3.20).

We are now in a position to establish the global well-posedness and the exponential decay estimates for solutions to (1.4) corresponding to small initial data.

Theorem 3.4

Assume that either Assumption 1.1 or Assumption 1.2 holds, together with Hypotheses 2.1 and 3.1. Let the initial data $U_{0} \in H$ satisfy

‖ U_{0} ‖_{H} < ρ,

(3.22)

for a sufficiently small constant

ρ > 0

. Then, the Cauchy problem (3.14) admits a unique global solution which decays exponentially in time according to estimate (3.20).

Proof.

Let us fix $T > 0$ , such that

{\tilde{M}}^{2} ρ^{2} e^{- (\tilde{α} - \tilde{M} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)}) T} < 1.

(3.23)

Moreover let

ρ > 0

be such that

ρ \leq \frac{1}{2 \sqrt{\tilde{C} (T)}} min_{i = 1, 2} {h_{i}^{- 1} (\frac{1}{2})} .

We consider initial data satisfying

‖ U_{0} ‖_{H} < ρ .

(3.24)

By Lemma 3.1, there exists a local solution

(u, y)

to (1.4) on some interval

[0, δ)

, with

δ > 0

. Without loss of generality, we may assume

δ < T

Using the monotonicity of $h_{i}$ and the fact that $\tilde{C} (T) > 1$ , we obtain

h_{1} (‖ \nabla u_{0} ‖_{L^{2} (Ω)}) \leq h_{1} (ρ) < \frac{1}{2}, h_{2} (‖ \nabla y_{0} ‖_{L^{2} (Ω)}) \leq h_{2} (ρ) < \frac{1}{2} .

Therefore, by Lemma 3.2, we have

\tilde{E} (0) > 0

. Moreover, from estimate (2.5), it follows that

\tilde{E} (0) \leq \frac{1}{2} ‖ u_{1} ‖_{L^{2} (Ω)}^{2} + \frac{3}{4} ‖ \nabla u_{0} ‖_{L^{2} (Ω)}^{2} + \frac{1}{2} ‖ y_{1} ‖_{L^{2} (Ω)}^{2} + \frac{3}{4} ‖ \nabla y_{0} ‖_{L^{2} (Ω)}^{2} \leq ρ^{2} .

(3.25)

This implies, for

i = 1, 2

h_{i} (2 \sqrt{\tilde{C} (T) \tilde{E} (0)}) < h_{i} (h_{i}^{- 1} (\frac{1}{2})) = \frac{1}{2} .

Hence, applying Lemma 3.3, we obtain the energy lower bound

\tilde{E} (t) > \frac{1}{4} ‖ u_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla u (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ y_{t} (t) ‖_{L^{2} (Ω)}^{2} + \frac{1}{4} ‖ \nabla y (t) ‖_{L^{2} (Ω)}^{2} > 0, \forall t \in [0, δ) .

(3.26)

From Proposition 3.1 and the fact that $δ < T$ , we deduce

\tilde{E} (t) \leq \tilde{C} (T) \tilde{E} (0), \forall t \in [0, δ) .

(3.27)

Combining (3.26) and (3.27), we obtain the bounds

\frac{1}{4} ‖ \nabla u (t) ‖_{L^{2} (Ω)}^{2} \leq \tilde{E} (t) \leq \tilde{C} (t) \tilde{E} (0), \frac{1}{4} ‖ \nabla y (t) ‖_{L^{2} (Ω)}^{2} \leq \tilde{E} (t) \leq \tilde{C} (t) \tilde{E} (0) .

Hence, the solution can be extended beyond

δ

up to

t = T

. In particular, for

t = T

, we have

h_{1} (‖ \nabla u (T) ‖_{L^{2} (Ω)}) < h_{1} (2 \sqrt{\tilde{C} (T) \tilde{E} (0)}) < \frac{1}{2}

and

h_{2} (‖ \nabla y (T) ‖_{L^{2} (Ω)}) < h_{2} (2 \sqrt{\tilde{C} (T) \tilde{E} (0)}) < \frac{1}{2} .

From (3.26) and (3.25), we also deduce

\frac{1}{4} ‖ U (t) ‖_{H}^{2} \leq \tilde{E} (t) \leq \tilde{C} (T) \tilde{E} (0) \leq \tilde{C} (T) ρ^{2}, \forall t \in [0, T],

which yields

‖ U (t) ‖_{H} \leq C_{ρ} := 2 \sqrt{\tilde{C} (T)} ρ, \forall t \in [0, T] .

Now, by possibly taking $ρ$ smaller, we can ensure that

L (C_{ρ}) < \frac{\tilde{α} - \tilde{M} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)}}{2 \tilde{M}} .

Hence, Theorem 3.3 implies that the solution satisfies the exponential decay estimate

‖ U (t) ‖_{H} \leq \tilde{M} ‖ U_{0} ‖_{H} e^{- \frac{\tilde{α} - \tilde{M} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)}}{2} t}, \forall t \in [0, T] .

(3.28)

In particular, we get

‖ U (t) ‖_{H}^{2} \leq {\tilde{M}}^{2} ρ^{2} e^{- (\tilde{α} - \tilde{M} ‖ a_{2}^{-} ‖_{L^{\infty} (Ω)}) t}, \forall t \in [0, T] .

Then,

‖ U (T) ‖_{H} \leq ρ,

where we have used (3.23).

Thus, we can iterate the argument on successive intervals $[T, 2 T]$ , $[2 T, 3 T]$ , and so on. By induction, we construct a unique global solution to (3.14). Furthermore, at each step, the exponential decay estimate (3.20) is preserved, ensuring that the global solution satisfies this decay property.

Footnotes

Acknowledgments

This work was started when AM was visiting the University of L’Aquila. AM thanks the MAECI (Ministry of Foreign Affairs and International Cooperation, Italy) for funding that greatly facilitated scientific collaborations between Moulay Ismail University of Meknes (Morocco) and University of L’Aquila (Italy).

ORCID iDs

Alhabib Moumni

Cristina Pignotti

Jawad Salhi

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: CP is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). She is partially supported by PRIN 2022 (2022238YY5) Optimal control problems: analysis, approximation and applications, PRIN-PNRR 2022 (P20225SP98) Some mathematical approaches to climate change and its impacts, and by INdAM GNAMPA Project Optimal Control and Machine Learning (CUP E5324001950001).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Adimy

Crauste

(2003). Global stability of a partial differential equation with distributed delay due to cellular replication. Nonlinear Analysis: Theory, Methods & Applications, 54, 1469–1491. 10.1016/S0362-546X(03)00197-4

Akil

Badawi

Nicaise

Wehbe

(2021). Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface. Mathematical Methods in the Applied Sciences, 44, 6950–6981. 10.1002/mma.7235

Akil

Badawi

Wehbe

(2021). Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay. Communications on Pure and Applied Analysis, 20, 2991–3028. 10.3934/cpaa.2021092

Alabau-Boussouira

(2002). Indirect boundary stabilization of weakly coupled hyperbolic systems. SIAM Journal on Control and Optimization, 41, 511–541. 10.1137/S0363012901385368

Alabau-Boussouira

(2003). A two level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. SIAM Journal on Control and Optimization, 42, 871–906. 10.1137/S0363012902402608

Alabau-Boussouira

(2013). A hierarchic multi-level energy method for the control of bidiagonal and mixed

n

-coupled cascade systems of PDEs by a reduced number of controls. Advances in Differential Equations, 18, 1005–1072. 10.57262/ade/1378327378

Alabau-Boussouira

Cannarsa

Sforza

(2008). Decay estimates for second order evolution equations with memory. Journal of Functional Analysis, 254, 1342–1372. 10.1016/j.jfa.2007.09.012

Alabau-Boussouira

Léautaud

(2013). Indirect controllability of locally coupled wave-type systems and applications. Journal de Mathematiques Pures et Appliquees, 99, 544–576. 10.1016/j.matpur.2012.09.012

Alabau-Boussouira

Wang

(2017). A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities. ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV), 23, 721–749. 10.1051/cocv/2016011

10.

Allaire

(2009). Analyse numérique et optimisation. Éditions de l’École Polytechnique.

11.

Ammari

Chentouf

Smaoui

(2022). Well-posedness and stability of a nonlinear time-delayed dispersive equation via the fixed point technique: A case study of no interior damping. Mathematical Methods in the Applied Sciences, 45, 4555–4566. 10.1002/mma.8052

12.

Ammari

Nicaise

Pignotti

(2010). Feedback boundary stabilization of wave equations with interior delay. Systems & Control Letters, 59, 623–628. 10.1016/j.sysconle.2010.07.007

13.

Avdonin

Rivero

A. C.

de Teresa

(2013). Exact boundary controllability of coupled hyperbolic equations. International Journal of Applied Mathematics and Computer Science, 23, 701–710. 10.2478/amcs-2013-0052

14.

Ayechi

Khenissi

Laurent

(2025). Indirect stabilization of semilinear coupled wave system. Mathematical Control and Related Fields (MCRF), 15, 493–514. 10.3934/mcrf.2024021

15.

Barbu

(1998). Partial differential equations and boundary value problems. Kluwer Academic Publishers.

16.

Bardos

Lebeau

Rauch

(1992). Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM Journal on Control and Optimization, 30, 1024–1065. 10.1137/0330055

17.

Benaddi

Rao

(2000). Energy decay rate of wave equations with indefinite damping. Journal of Differential Equations, 161, 337–357. 10.1006/jdeq.2000.3714

18.

Bennour

Ammar Khodja

Teniou

(2017). Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations and Control Theory, 6, 487–516. 10.3934/eect.2017025

19.

Berrimi

Messaoudi

S. A.

(2006). Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Analysis: Theory, Methods & Applications, 64, 2314–2331. 10.1016/j.na.2005.08.015

20.

Cazenave

Haraux

(1998). An introduction to semilinear evolution equations. Oxford University Press.

21.

Chellaoua

Boukhatem

Feng

(2023). Well-posedness and stability for an abstract evolution equation with history memory and time delay in hilbert space. Advances in Differential Equations, 28, 953–980. 10.57262/ade028-1112-953

22.

Continelli

Pignotti

(2025). Energy decay for semilinear evolution equations with memory and time-dependent time delay feedback. Advances in Differential Equations, 30, 601–634. 10.57262/ade030-0910-601

23.

Coron

J.-M.

(2007). Control and nonlinearity. American Mathematical Society.

24.

Datko

(1991). Two questions concerning the boundary control of certain elastic systems. Journal of Differential Equations, 92, 27–44. 10.1016/0022-0396(91)90062-E

25.

Fragnelli

Pignotti

(2016). Stability of solutions to nonlinear wave equations with switching time delay. Dynamics of Partial Differential Equations, 13, 31–51. 10.4310/DPDE.2016.v13.n1.a2

26.

Freitas

Zuazua

(1996). Stability results for the wave equation with indefinite damping. Journal of Differential Equations, 132, 338–352. 10.1006/jdeq.1996.0183

27.

Gerbi

Kassem

Mortada

Wehbe

(2021). Exact controllability and stabilization of locally coupled wave equations: Theoretical results. Zeitschrift für Analysis und ihre Anwendungen, 40, 67–96. 10.4171/zaa/1673

28.

Haraux

(1989). Une remarque sur la stabilisation de certains systemes du deuxieme ordre en temps. Portugaliae Mathematica, 46, 245–258. https://eudml.org/doc/115668

29.

Koumaiha

Toufaily

Wehbe

(2022). Boundary observability and exact controllability of strongly coupled wave equations. Discrete & Continuous Dynamical Systems-Series S, 15, 1269–1305. 10.3934/dcdss.2021091

30.

Liu

(1997). Locally distributed control and damping for conservative systems. SIAM Journal on Control and Optimization, 35, 1574–1590. 10.1137/S0363012995284928

31.

Liu

Rao

(2009). A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems, 23, 399–414. 10.3934/dcds.2009.23.399

32.

López-Gómez

(1997). On the linear damped wave equation. Journal of Differential Equations, 134, 26–45. 10.1006/jdeq.1996.3209

33.

Luo

J. R.

Xiao

T. J.

(2023). Optimal decay rates for semi-linear non-autonomous evolution equations with vanishing damping. Nonlinear Analysis: Theory, Methods & Applications, 230. Paper No. 113247. 10.1016/j.na.2023.113247

34.

Menz

(2007). Exponential stability of wave equations with potential and indefinite damping. Journal of Differential Equations, 242, 171–191. 10.1016/j.jde.2007.04.002

35.

Mokhtari

Ammar Khodja

(2022). Boundary controllability of two coupled wave equations with space-time first-order coupling in 1-D. Journal of Evolution Equations, 22. Paper No. 31. 10.1007/s00028-022-00790-x

36.

Moumni

Mehdaoui

Salhi

Tilioua

(2024). Theoretical and numerical indirect stabilization of coupled wave equations with a single time-delayed damping. arXiv:2410.09995.

37.

Nicaise

Pignotti

(2006). Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM Journal on Control and Optimization, 45, 1561–1585. 10.1137/060648891

38.

Oliveira

R. L.

Oquendo

H. P.

(2020). Stability and instability results for coupled waves with delay term. Journal of Mathematical Physics, 61, 071505. 10.1063/1.5144987

39.

Paolucci

Pignotti

(2022). Well-posedness and stability for semilinear wave-type equations with time delay. Discrete & Continuous Dynamical Systems-Series S, 15, 1561–1571. 10.3934/dcdss.2022049

40.

Pazy

(2012). Semigroups of linear operators and applications to partial differential equations. Springer.

41.

Pignotti

(2012). A note on stabilization of locally damped wave equations with time delay. Systems & Control Letters, 61, 92–97. 10.1016/j.sysconle.2011.09.016

42.

Pignotti

(2024). Exponential decay estimates for semilinear wave-type equations with time-dependent time delay. arXiv:2303.14208.

43.

Rebiai

S. E.

Sidi Ali

F. Z.

(2014). Exponential stability of compactly coupled wave equations with delay terms in the boundary feedbacks. In System Modeling and Optimization: 26th IFIP TC 7 Conference, CSMO 2013, Klagenfurt, Austria, September 9–13, 2013, Revised Selected Papers (pp. 278–284). Springer.

44.

Silga

Kyelem

B. A.

Bayili

(2022). Indirect boundary stabilization with distributed delay of coupled multi-dimensional wave equations. Annals of the University of Craiova-Mathematics and Computer Science Series, 49, 15–34. 10.52846/ami.v49i1.1430

45.

Wehbe

Youssef

(2011). Indirect locally internal observability and controllability of weakly coupled wave equations. Difference Equations and Applications, 3, 449–462.

46.

G. Q.

Yung

S. P.

L. K.

(2006). Stabilization of wave systems with input delay in the boundary control. ESAIM: Control, Optimisation and Calculus of Variations, 12, 770–785. 10.1051/cocv:2006021

47.

G. Q.

Zhang

(2020). Uniform stabilization of 1-D coupled wave equations with anti-dampings and joint delayed control. SIAM Journal on Control and Optimization, 58, 3161–3184. 10.1137/19M1289145

48.

Zuazua

(1990). Exponential decay for the semilinear wave equation with locally distributed damping. Communications in Partial Differential Equations, 15, 205–235. 10.1080/03605309908820684