Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights

1.   Introduction

This paper is concerned with the initial boundary value problem

where $\Omega = ]0,L[$, $0<\tau(t)$ are a non-constant time delay, $\mu_1(t), \mu_2(t)$ are non-constant weights and the initial data $(u_0,u_1,f_0)$ belong to a suitable function space.

This problem has been first proposed and studied in Nicaise and Pignotti [22] in case of constant coefficients $\mu_1, \mu_2$ and constant time delay. Under suitable assumptions, the authors proved the exponential stability of the solution by introducing suitable energies and by using some observability inequalities. Some instability results are also given for the case of the some assumptions is not satisfied.

With a weight depending on time, $\mu_1(t), \mu_2(t)$ and constant time delay, this problem was studied in [2], where the existence of solution was made by Faedo-Galerkin method and a decay rate estimate for the energy was given by using the multiplier method.

W. Liu in [19] studied the weak viscoelastic equation with an internal time varying delay term. By introducing suitable energy and Lyapunov functionals, he establishes a general decay rate estimate for the energy under suitable assumptions.

F. Tahamtani and A. Peyravi [29] investigated the nonlinear viscoelastic wave equation with source term. Using the Potential well theory they showed that the solutions blow up in finite time under some restrictions on initial data and for arbitrary initial energy.

Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term was considered by M. Remil and A. Hakem in [28].

Global existence and asymptotic stability for a coupled viscoelastic wave equation with time-varying delay was studied in [3] by combining the energy method with the Faedo-Galerkin's procedure.

The stabilization problem by interior damping of the wave equation with boundary or internal time-varying delay was studied in [23] by introducing suitable Lyapunov functionals.

Energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks was considered in [11].

For problems with delay in different contexts we cite [9,10,30,32] with references therein. In absence of delay ($\mu_2(t) = 0$), the problem (1) is exponentially stable provided that $\mu_1(t)$ is constant, see, for instance [5,6,16,17,21] and reference therein.

Time delay is the property of a physical system by which the response to an applied force is delayed in its effect, and the central question is that delays source can destabilize a system that is asymptotically stable in the absence of delays, see [7]. In fact, an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay unless additional control terms have been used, see for example [8,12,31]

By energy method in [24] was studied the stabilization of the wave equation with boundary or internal distributed delay. By semigroup approach in [27] was proved the well-posedness and exponential stability for a wave equation with frictional damping and nonlocal time-delayed condition. Transmission problem with distributed delay was studied in [18] where was established the exponential stability of the solution by introducing a suitable Lyapunov functional.

Here we consider a wave equation with non-constant delay and nonlinear weights, thus, the present paper is a generalization of the previous ones. The remaining part of this paper is organized as follows. In the section 2 we introduce some notations and prove the dissipative property of the full energy of the system. In the section 3, for an approach combining semigroup theory (see [21] and [4]) with the energy estimate method we prove the existence and uniqueness of solution. In section 4 we present the result of exponential stability.

2.   Notation and preliminaries

We will need the following hypotheses:

(H1) $\mu_1:\mathbb{R}_+ \rightarrow ]0,+\infty[$ is a non-increasing function of class $C^1(\mathbb{R}_+)$ satisfying

where $\alpha_0$ and $M_1$ are constants such that $M_1>0$.

(H2) $\mu_2:\mathbb{R}_+ \rightarrow \mathbb{R}$ is a function of class $C^1(\mathbb{R}_+)$, which is not necessarily positives or monotones, such that

for some $0 < \beta < \sqrt{1-d}$ and $M_2>0$.

We now state a lemma needed later.

Lemma 2.1 (Sobolev-Poincare's inequality). Let $q$ be a number with $2\leq q \leq +\infty$. Then there is a constant $c_* = c_*(]0,L[,q)$ such that

Lemma 2.2 ([13][16]). Let $E:\mathbb{R}_+ \rightarrow \mathbb{R}_+$ be a non increasing function and assume that there are two constants $\sigma>-1$ and $\omega>0$ such that

Then

As in [23], we assume that

and there exist positive constants $\tau_0,\tau_1$ and $d$ satisfying

and

We introduce the new variable

Then

and problem (1) takes the form

We define the energy of the solution of problem (9) by

where

is a non-increasing function of class $C^1(\mathbb{R}_+)$ and $\bar{\xi}$ be a positive constant such that

Our first result states that the energy is a non-increasing function.

Lemma 2.3. Let $(u,z)$ be a solution to the problem (9). Then, the energy functional defined by (10) satisfies

Proof. Multiplying the first equation (9) by $u_t(x,t)$, integrating on $\Omega$ and using integration by parts, we get

Now multiplying the second equation (9) by $\xi(t)z(x,\rho,t)$ and integrate on $\Omega \times ]0,1[$, to obtain

Consequently,

From (10), (14) and (15) we obtain

Due to Young's inequality, we have

Inserting (17) into (16), we obtain

Lemma 2.4. Let $(u,z)$ be a solution to the problem (9). Then the energy functional defined by (10) satisfies

where $\sigma = a_0\left( 1-\frac{\bar{\xi}}{2}- \frac{\beta}{2\sqrt{1-d}} \right)$.

Proof. From Lemma 2.3, we have that

and from (H1), we obtain

and the lemma is proved.

3.   Global solution

For the semigroup setup we $U = (u,u_t,z)^T$ and rewrite (9) as

where the operator $\mathcal{A}(t)$ is defined by

We introduce the phase space

and the domain of $\mathcal{A}$ is defined by

where

Notice that the domain of the operator $\mathcal{A}(t)$ is independent of the time $t$, i.e.,

$\mathcal{H}$ is a Hilbert space provided with the inner product

for $U = (u,v,z)^T$ and $\bar{U} = (\bar{u},\bar{v},\bar{z})^T$.

Using this time-dependent inner product and the next theorem we will get a result of existence and uniqueness.

Theorem 3.1. Assume that

(i) $Y = D(\mathcal{A}(0))$ is dense subset of $\mathcal{H}$,

(ii) (21) holds,

(iii) for all $t \in [0,T]$, $\mathcal{A}(t)$ generates a strongly continuous semigroup on $\mathcal{H}$ and the family $\mathcal{A}(t) = \left\{ \mathcal{A}(t)/ t \in [0,T] \right\}$ is stable with stability constants $C$ and $m$ independent of $t$ (i.e., the semigroup $(S_t(s))_{s\geq 0}$ generated by $\mathcal{A}(t)$ satisfies $\| S_t(s)u \|_{\mathcal{H}} \leq Ce^{ms} \| u \|_{\mathcal{H}}$, for all $u \in \mathcal{H}$ and $s\geq 0$),

(iv) $\partial_t \mathcal{A}(t)$ belongs to $L_{*}^{\infty}([0,T],B(Y, \mathcal{H}))$, which is the space of equivalent classes of essentially bounded, strongly measurable functions from $[0,T]$ into the set $B(Y, \mathcal{H})$ of bounded operators from $Y$ into $\mathcal{H}$.

Then, problem (18) has a solution $U \in C([0,T],Y) \cap C^1([0,T], \mathcal{H})$ for any initial datum in $Y$.

Our goal is then to check the above assumptions for problem (18).

First, we prove $D(\mathcal{A}(0))$ is dense in $\mathcal{H}$.

The proof is the same as the one Lemma $2.2$ of [25], we give it for the sake of completeness.

Let $(f,g,h)^T$ be orthogonal to all elements of $D\mathcal{A}(0)$, namely

for all $(u,v,z)^T \in D(\mathcal{A}(0))$.

We first take $u = v = 0$ and $z \in D(\Omega \times ]0,1[)$. As $(0,0,z)^T \in D(\mathcal{A}(0))$, we get

Since $D(\Omega \times ]0,1[)$ is dense in $L^2(\Omega \times ]0,1[)$, we deduce that $h = 0$. In the same manner, by taking $u = z = 0$ e $v \in D(\Omega)$ we see that $g = 0$.

The above orthogonality condition is then reduced to

By restricting ourselves to $v = 0$ and $z = 0$, we obtain

Since $D(\Omega)$ is dense in $H_0^1(\Omega)$ (equipped with the inner product $\langle \cdot, \cdot \rangle_{H_0^1(\Omega}$), we deduce that $f = 0$.

We consequently

Secondly, we notice that

where $\Phi = (u,v,z)^T$ and $c$ is a positive constant and $\| \cdot \|$ is the norm associated the inner product (22). For all $t,s \in [0,T]$, we have

It is clear that $1 - e^{\frac{c}{\tau_0}|t-s|} \leq 0$. Now we will prove $\xi(t)\tau(t) - \xi(s)\tau(s)e^{\frac{c}{\tau_0}|t-s|} \leq 0$ for some $c>0$. To do this, we have

where $r \in ]s,t[$.

Hence $\xi$ is a non increasing function and $\xi>0$, we get

which implies

Using (5) and $\tau'$ is bounded, we deduce that

which proves (24) and therefore $(iii)$ follows.

Now we calculate $\langle \mathcal{A}(t)U,U \rangle_t$ for a fixed $t$. Take $U = (u,v,z)^T \in D(\mathcal{A}(t))$. Then

Integrating by parts, we obtain

Since

we have

So we get

Therefore, from (16) and (17), we deduce

Then, we have

where

From the (13), we obtain

which means that the operator $\tilde{\mathcal{A}} = \mathcal{A}(t) - \kappa(t)I$ is dissipative.

Moreover, $\kappa'(t) = \frac{\tau'(t)\tau''(t)}{2\tau(t)\sqrt{1+\tau'(t)^2}} - \frac{\tau'(t)\sqrt{1+\tau'(t)^2}}{2\tau(t)^2}$ is bounded on $[0,T]$ for all $T>0$ (by (5) and (12)) and we have

with $\frac{\tau''(t)\tau(t)\rho-\tau'(t)(\tau'(t)\rho-1)}{\tau(t)^2}$ bounded on $[0,T]$ by (5) and (12). Thus

the space of equivalence classes of essentially bounded, strongly measurable functions from $[0,T]$ into $B(D(\mathcal{A}(0)), \mathcal{H})$.

Now, we will show that $\lambda I - \mathcal{A}(t)$ is surjective for fixed $t>0$ and $\lambda > 0$. For this purpose, let $F = (f_1,f_2,f_3)^T \in \mathcal{H}$, we seek $U = (u,v,z)^T \in D(\mathcal{A}(t))$ solution of

that is verifying following system of equations

Suppose that we have found $u$ with the appropriated regularity. Then

It is clear that $v \in H_0^1(\Omega)$. Furthermore, by (27) we can find $z$. From (20), we have

Following the same approach as in [22], we obtain, by using equation for $z$ in (27),

if $\tau'(t) = 0$, where $\vartheta(\ell,t) = \lambda \ell \tau(t)$, and

otherwise, where $\zeta(\ell,t) = \lambda \frac{\tau(t)}{\tau'(t)}\ln(1- \ell \tau'(t))$.

From (28), we obtain

if $\tau'(t) = 0$, and

otherwise.

In particular, if $\tau'(t) = 0$ and from (30), we have

and if $\tau'(t) \neq 0$ and from (31), we have

By using (27) and (28), the function $u$ satisfies

Solving the equation (34) is equivalent to finding $u \in H^2(\Omega) \cap H_0^1(\Omega)$ such that

for all $\eta \in H_0^1(\Omega)$.

Consequently, the equation (35) is equivalent to the problem

where the bilinear form

and the linear form

are defined by

and

where

and

It is easy to verify that $\Upsilon$ is continuous and coercive, and $L$ is continuous. So applying the Lax-Milgram theorem, we deduce that for all $\eta \in H_0^1(\Omega)$ the problem (36) admits a unique solution

Applying the classical elliptic regularity, it follows from (35) that

Therefore, the operator $\lambda I - \mathcal{A}(t)$ is surjective for any $\lambda > 0$ and $t>0$. Again as $\kappa(t)>0$, this prove that

for any $\lambda >0$ and $t>0$.

Then, (24), (25) and (37) imply that the family $\tilde{\mathcal{A}} = \left\{ \tilde{\mathcal{A}}(t)/ t \in [0,T] \right\}$ is a stable family of generators in $\mathcal{H}$ with stability constants independent of $t$, by Proposition $1.1$ from [14]. Therefore, the assumptions $(i)-(iv)$ of Theorem 3.1 are verified by (21), (24), (25), (26), (37) and (23), and thus, the problem

has a unique solution $\tilde{U} \in C\left( [0,+\infty[, D(\mathcal{A}(0)) \right) \cap C^1\left( [0,+\infty[, \mathcal{H} \right)$ for $U_0 \in D(\mathcal{A}(0))$. The requested solution of (18) is then given by

because

which concludes the proof.

The existence and uniqueness are obtained by the following result.

Theorem 3.2 (Global solution). For any initial datum $U_0 \in \mathcal{H}$ there exists a unique solution $U$ satisfying

for problem (18).

Moreover, if $U_0 \in D(\mathcal{A}(0))$, then

Proof. A general theory for equations of type (18) has been developed using semigroup theory [14], [15] and [26]. The simplest way to prove existence and uniqueness results in to show that the triplet $\left\{ (\mathcal{A}, \mathcal{H}, Y) \right\}$, with $\mathcal{A} = \left\{ \mathcal{A}(t)/ t \in [0,T] \right\}$, for some fixed $T>0$ and $Y = \mathcal{A}(0)$, forms a CD-systems (or constant domain system, see [14] and [15]). More precisely, the following theorem gives the existence and uniqueness results and is proved in Theorem $1.9$ of [14] (see also Theorem $2.13$ of [15] or [1]).

4.   Asymptotic behavior

In this section we shall investigate the asymptotic behavior of problem (1). The stability result will be obtained using Lemma 2.2.

Theorem 4.1 (Stability Result). Let $(u_0,u_1,f_0(\cdot,-,\tau(0))) \in H_0^1(\Omega) \times L^2(\Omega) \times L^2(\Omega \times ]0,1[)$. Assume that the hypotheses (H1), (H2) and (5)-(7) hold. Then problem (1) admits a unique solution

Proof. From now on, we denote by $c$ various positive constants which may be different at different occurrences.

Given $0 \leq S < T < \infty$ we start by multiplying the first equation of (9) by $uE^q$ and then integrating over $(S,T) \times \Omega$, we obtain

Notice that

using integration by parts and the boundary conditions we know that

Similarly, we multiply the second equation of (9) by $E^q \xi(t) e^{-2\rho\tau(t)}z(x,\rho,t)$ and then integrate over $\Omega \times (0,1) \times (S,T)$ to see that

Using integration by parts and the boundary conditions we know that

Since $\mu_1$ is a non-increasing function of class $C_1(\mathbb{R})$, its derivatives is non-positive, which implies that $\xi'(t) \leq 0$. This result this

Moreover, as

then, from (40), (41) and (42), we have that

Using the definition of $E$, (39) and (43), we get

where $\gamma_0 = 2\min \{ 1, e^{-2\tau_1} \}$.

Using the Young and Sobolev-Poincaré inequalities and Lemma 2.3, we find that

Now, we known that

By (13), we have

Similarly,

From Lemma 2.4, we deduce that

Now, we get that

and from (H2) we obtain that

Finally,

Choosing $\varepsilon_1$ and $\varepsilon_2$ small enough, we deduce from (45) and (46) that

Since $E(S) \leq E(0)$ for $S \geq 0$, we have that

We choose $q = 0$, we conclude from Lemma 2.2 that

This ends the proof of Theorem 4.1.