Existence and energy decay rate of the solutions for the wave equation with a nonlinear distributed delay

1.   Introduction

In this paper, we consider the following wave equation with nonlinear distributed delay:

where $\Omega \subset \mathbb{R}^n$, $n \geq 1$ is a bounded domain with smooth boundary $\partial\Omega$, $\mu(s) \in L^{\infty}([\tau_{1}, \tau_{2}]; \mathbb{R}_+)$ with $\tau_{2} > \tau_{1} > 0$, and $\mu_{0}$ is for some positive constant that will be specified later.

Time delay effect occurs in many various phenomena depending on past states as well as on present situations, so time delay problem is widely applied in many engineering and biology fields ^[1,21]. Hence the partial differential equations with such circumstance have been studied by many researchers (see ^{[3,4,10,11,12,13,14,15,16,22,23]} and a list on references therein). For example, Nicaise and Pignotti ^[13] considered the wave equation with a delay concentrated at a time

They proved that this equation is either exponentially stable under condition $\mu_{0} > \mu_{1}$ or unstable under condition $\mu_{0}\le\mu_{1}$. Benaissa et al. ^[4] studied the global existence and energy decay of solutions to a viscoelastic wave equation with delay term in the nonlinear internal feedback. They proved the global existence result using Galerkin's method and the asymptotic behavior of solutions using a perturbed energy method. While there are many results dealing with a delay concentrated at a time, there are relatively few researches dealing with a distributed delay. The distributed delay is important and has been studied in many problems (see ^{[6,7,8,14,19]}). For instance, Nicaise and Pignotti ^[14] has studied the wave equation with linear distributed delay

They proved the global well-posedness by the semigroup theory and the exponential stability under the assumption

Raposo et al. ^[19] proved the well-posedness using the semigroup theory and the exponential stability exploiting the dissipative properties of the linear operator associated to damped model using the Gearhart-Huang-Pruss theorem for the wave equation with frictional damping and nonlocal time-delayed condition. Recently, Choucha et al. ^[7] studied a coupled Lame system with distributed delay, viscoelastic, and logarithmic source terms. They proved an exponential decay of solutions by using Lyapunov functional method. But, the above mentioned references were considered a linear distributed delay. There is none, as far as we know, well-posedness result dealing with a nonlinear distributed delay.

Motivated by previous works, the goal of the paper is to study the existence and energy decay of the solutions for the wave equation with a nonlinear distributed delay. We prove the local existence of the solutions by using the semigroup theory, where the source term is globally Lipschitz and then establish the global existence of solutions and the energy decay result under the local Lipschitz source and suitable conditions on the initial data.

Throughout this paper, we use standard functional spaces and $L^p(\Omega)$-norm is denoted by $\|\cdot\|_p$, and $(u, v) = \int_\Omega u(x)v(x) dx$. The following assumptions are made on the nonlinear functions $f$ and $g$.

(A1) $f : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous monotone increasing function with $f(0) = 0$.

(A2) There exist positive constants $\nu$, $M_1$, $M_2$ and a convex increasing function $H:\mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ of the class $H \in C^{1}(\mathbb{R}_{+}) \cap C^{2}((0, \infty))$ satisfying $H(0) = 0$, and $H$ is linear in $[0, \nu]$ or $H'(0) = 0$ and $H'' > 0$ on $(0, \nu]$ such that

(A3) $g$ is an odd nondecreasing Lipschitz function.

(A4) $\alpha_{1}sg(s) \le G(s) \le \alpha_{2}sf(s)$, where $\alpha_{1}, \alpha_{2}$ are some positive constants, and $G(s) = \int_{0}^{s} g(r) dr$.

2.   Well-posedness

2.1.   Globally Lipschitz source

We first deal with the case where the source $h$ is globally Lipschitz from $H^{1}_0(\Omega)$ into $L^{2}(\Omega)$. We will prove the problem (1.1) are well-posed using the semigroup theory.

Let us set $u_{t}(x, t-\rho s): = y(x, \rho, t, s)$, $\rho \in (0, 1)$, $s \in (\tau_{1}, \tau_{2})$. Then the problem (1.1) is transformed into

If we set $u_{t}: = v.$ and $U: = (u, v, y)^{T}$, then (2.1) can be rewritten as

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

with $D(\mathcal{A}): = \{(u, v, y)^{T} \in (H^{2}(\Omega)\cap H_{0}^{1}) \times H_{0}^{1}(\Omega) \times L^{2}(\Omega \times (\tau_{1}, \tau_{2});$ $H^1(0, 1)) : v(x) = y(x, 0, s) \quad \text{in} \quad \Omega \}$. So we see that in order to obtain the existence of the solutions to the problem (1.1), it is sufficient to show that the problem (2.2) admits a solution.

We define the Hilbert space $\mathcal{H}$

with inner product

Theorem 2.1. Assume that (A1) and (A3) hold. In addition, $h$ is globally Lipschitz from $H^{1}_0(\Omega)$ into $L^{2}(\Omega)$. Then (2.2) has a unique solution

for $(u_{0}, v_{0}, f_{0})^{T} \in \mathcal{H}$.

Proof. First, we will show that the operator $\mathcal{A}$ is $w$-accretive on $\mathcal{H}$. Let $U = (\xi, \eta, y)^{T}, V = (\tilde{\xi}, \tilde{\eta}, \tilde{y})^{T} \in D(\mathcal{A})$. Then we get

Using Hölder's and Young's inequalities, we obtain for sufficiently large $w$ and small $\varepsilon$,

where $L_{h}$ and $L_{g}$ are Lipschitz constants for $h$ and $g$, respectively. Thus $\mathcal{A}$ is $w$-accretive.

Next, we will show that $\mathcal{A} + wI$ is a maximal monotone operator. To this end, it is sufficient to show that $R(\lambda I+\mathcal{A}) = \mathcal{H}$ for sufficiently large constant $\lambda$.

Given $(k, l, m) \in \mathcal{H}$, We seek $U = (\xi, \eta, y)^{T} \in D(\mathcal{A})$ satisfying

From (2.3) and (2.5) with $y(x, 0, s) = \eta(x)$, we have $y(x, 1, s) = \eta(x) e^{-\lambda s}+X_{0}$, where

Hence, by (2.4) we obtain

We will show that $T:H^{1}_0(\Omega) \rightarrow H^{-1}(\Omega)$ is surjective. Let $\hat{\eta}: = \eta e^{-\lambda s}+X_{0}$, and let

It easy to see that $\mathcal{B}$ is maximal monotone. From the fact that $g$ and $h$ are global Lipschitz, and $\lambda I-\frac{1}{\lambda}\Delta$ is continuous and coercive, we infer that, for large constant $\lambda > 0$,

where $c$ is positive constant. Therefore $\mathcal{C}$ is maximal monotone operator and coercive. Thus $\mathcal{B} + \mathcal{C}$ is maximal monotone, which implies that $T$ is surjective. So we obtain

Thus we have $\left(\xi, \eta, y\right)^{T} \in D(\mathcal{A})$, consequently, the operator $\mathcal{A} + wI$ is maximal monotone.

From the fact the density of $D(\mathcal{A})$ in $\mathcal{H}$ and the nonlinear semigroup theory ^[2,20], the proof of Theorem 2.1 is completed.

2.2.   Locally Lipschitz source

In this subsection, we loosen the globally Lipschitz condition on the source by allowing $h$ to be locally Lipschitz continuous. We first introduce the Legendre transformation. Let $G^{*}$ be conjugate of convex function $G$. It is defined by $G^{*}(s) = \sup_{t\in \mathbb{R}_{+}}\left(st-G(t)\right)$. $G^{*}$ is called Legendre transform of $G$. By definition, we have

and

By the assumption (A4) and (2.6), we obtain

Let us define functional

where

To estimate this subsection, we need the following additional assumption:

Theorem 2.2. Assume that (A1), (A3), (A4), (2.10) and (2.11) hold, and let $h:H^{1}_0(\Omega) \rightarrow L^{2}(\Omega)$ be local Lipschitz continuous function. Then (1.1) has unique local solution for $(u_{0}, v_{0}, f_{0})^{T} \in \mathcal{H}$ such that

Proof. Define

where $k$ is a positive constant. With this truncated function $h_k$, we consider the following problem:

Since $h_{k}:H^{1}_0\left(\Omega \right) \rightarrow L^{2}\left(\Omega\right)$ is global Lipschitz with Lipschitz constant $L_{h_{k}}$ for each $k$ (see ^[9]), by Theorem 2.1, the problem (2.12) has a unique solution $u_k \in C(0, \infty; H_{0}^{1}(\Omega)) \cap C^{1}(0, \infty; L^{2}(\Omega))$. To simplify the notation in the rest of the proof, we shall express $u_k$ as $u$. We will use the following notation: $\int_{\mathrm{Q}} = \int_{0}^{T}\int_{\Omega}, \ d\mathrm{Q} = d\Omega dt.$

Multiplying (2.12) by $u_{t}$, we obtain

Since

we have

Integrating (2.13) over (0, T) and using (2.9), we get

By using (2.7) and (2.8), we deduce that

From the assumptions (2.10), (2.11), and using the Young inequality, (2.14) is rewritten as

Hence, by Gronwall's inequality, we obtain

If we choose $T$ such that $2\xi(0)+2C_{h_{k}}T < k^{2}$, there exists

such that $\Vert u(t) \Vert_{H^{1}} < k$ for all $t < T_{k}$. Hence the original problem same as problem (2.12) for $t < T_{k}$. By repeating the same process with initial data $u(T_{k})$ and for large $k$, we have maximal time $T_{max}$. Thus the proof of Theorem 2.2 is completed.

3.   Asymptotic behavior

In this section, we prove the global existence and energy decay of the solutions to the problem (1.1) when

Since $h : H^{1}_0(\Omega) \rightarrow L^{2}(\Omega)$ is locally Lipschitz, Theorem 2.2 allows of this polynomial growth source. In the following section, the symbol $C$ is a generic positive constant, which may be different in various occurrences. The energy associated to the problem (1.1) is defined by

Then from (2.13) and (2.14), we have

which implies that $E(t)$ is a nonincreasing function.

We now set

and

Then we know that ^[18]

where

From the relationship $J(u) = \frac{1}{2}I(u) +\frac{p-2}{2p}\Vert u \Vert_{p}^{p}$, the energy $E(t)$ is rewritten as

Lemma 3.1. If $E(0) < d$ and $I(u_{0}) > 0$, then $I(u(t)) > 0$ for all $t \in [0, T_{max})$.

Proof. The proof is same as that of [17, Lemma 4.1], so we omit it here.

Theorem 3.1. Under the assumptions on Theorem 2.2 and Lemma 3.1, (1.1) has a unique global solution for $(u_{0}, v_{0}, f_{0})^T \in \mathcal{H}$.

Proof. It suffices to show that $\Vert u_t \Vert _{2}^{2}+\Vert \nabla u \Vert_{2}^{2}$ is bounded independent of $t$. By Lemma 3.1 and (3.2), we get

and

Under the assumption Lemma 3.1 and by definition of energy, we easily obtain

where $\epsilon$ is a for some sufficiently small positive constant. Now we recall the technical lemma which will play an essential role when establishing the energy decay.

Lemma 3.2. (see ^[5]) Let $E:\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ be a non-increasing differentiable function and $\Psi:\mathbb{R_{+}}\rightarrow \mathbb{R_{+}}$ be a convex and increasing function s.t $\Psi(0) = 0$. Assume that

Then $E$ satisfies the following estimate:

where $\psi(t) = \int_{t}^{1}\frac{1}{\Psi(s)}ds$ for $t > 0, h(t) = 0$ for $0\le t \le \frac{E(0)}{\Psi(E(0))},$ and

Thanks to ^[3] idea, we obtain following theorem.

Theorem 3.2. Under the assumptions on Theorem 3.1, we obtain the following energy decay property:

where $\psi(t) = \int_{t}^{1}\frac{1}{\omega\varphi(s)}ds$ for $t > 0, h(t) = 0$ for $0\le t \le \frac{E(0)}{\omega\varphi(E(0))},$ and

for some positive constant $\omega$, and

for some positive constant $\epsilon_0$.

Proof. We multiply the first equation of (2.1) by $A\frac{\varphi(E)}{E}u$ where $A = e^{-2\tau_{2}}$, and then integrate the obtained result over $(S, T) \times \Omega$. Then we have

Similarly, multiplying the second equation of (2.1) by $\xi_{0}\frac{\varphi(E)}{E}e^{-2s\rho}g(y(x, \rho, t, s))$, and integrating over $(S, T) \times \Omega \times (\tau_{1}, \tau_{2}) \times (0, 1)$, we obtain

Combining the above two equations, we get the following equation:

Now we are going to estimate terms on the right hand side of (3.4).

From (3.2) and (3.3), we have

We know by assumption of $\varphi(s)$, $\frac{\varphi(s)}{s}$ is nondecreasing and $E$ is nonnegative and decreasing. Also by using Young's inequality and Poincaré inequality, we obtain

Since $g$ is a nondecreasing odd Lipschitz function and $g^{2}(y(x, 1, t, s) \leqslant L_{g}g(y(x, 1, t, s))y(x, 1, t, s)$, we have

By assumption $(A4)$ and (3.1), we obtain

Replacing (3.5)–(3.12) in (3.4) and taking $\epsilon$ sufficiently small, we arrive at

We are now going to estimate the last two terms of right-hand side of (3.13). We consider the two cases with respect to the conditions on $H$.

Case 1. $H$ is linear on $[0, \nu]$.

From the assumption (A2) and the definition of the energy $E(t)$, we have

Put

By assumption $(A1), (A2)$ and using (3.1), we have

Case 2. $H'(0) = 0, H'' > 0 \ on\ (0, \nu]$. By the assumption (A2), we obtain

On the other hand, by applying Jensen's inequality for concave function we deduce that

By using Legendre transform (2.6) and (2.7) for $H$ and $\varphi(s): = sH'(\varepsilon_{0}s)$, we obtain

Therefore, choosing $\varepsilon_{0}$ small enough we get in both cases

Thus, applying Lemma 3.2 with $\Psi(t) = \omega\varphi(t)$, the proof of Theorem 3.2 is completed.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2022R1I1A3055309).

Conflict of interest

The authors declare no conflict of interest.