Existence and Uniqueness of a Solution to a Wentzell’s Problem with Non-Linear Delays ^†

Abstract

In this work, we study the existence and uniqueness of the solution to a wave equation with dynamic Wentztell-type boundary conditions on a part of the boundary ${\Gamma }_1$ of the domain $\Omega$ with non-linear delays in non-linear dampings in $\Omega$ and on ${\Gamma }_1$, using the Faedo–Galerkin method.

1. Introduction

We consider the following coupled system wave/Wentzell:

$$\left\{\begin{array}{c}{u}_{tt}-\Delta u+{\mu }_1{g}_1\left(u_t\right)+{\mu }_2{g}_1\left(u_t(t-\tau )\right)=0,\mathrm{in}\Omega \times (0,\infty ),\\ v_{tt}+{\partial }_{\nu }u-{\Delta }_{T}v+{\mu }_1^{\prime }{g}_2\left(v_t\right)+{\mu }_2^{\prime }{g}_2\left(v_t(t-\tau )\right)=0,\mathrm{on}{\Gamma }_1\times (0,\infty ),\\ u=v,\mathrm{on}\Gamma \times (0,\infty ),\\ u=0,\mathrm{on}{\Gamma }_0\times (0,\infty ),\\ (u\left(0\right),v\left(0\right))=(u_0,v_0),\mathrm{in}\Omega \times \Gamma ,\\ (u_t\left(0\right),v_t\left(0\right))=(u_1,v_1),\mathrm{in}\Omega \times \Gamma ,\\ u_t(x,t-\tau )=f_{0_1}(x,t-\tau ),\mathrm{in}\Omega \times (0,\tau ),\\ v_t(x,t-\tau )=f_{0_2}(x,t-\tau ),\mathrm{on}{\Gamma }_1\times (0,\tau ),\end{array}\right.$$

(1)

where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$, $(n\ge 2),$ with smooth boundary $\Gamma =\partial \Omega ,$ divided into two closed and disjoint subsets ${\Gamma }_0$ and ${\Gamma }_1,$ such that $\overline{{\Gamma }_0}\cap \overline{{\Gamma }_1}=\varnothing$ and ${\Gamma }_0\cup {\Gamma }_1=\Gamma$. We denote by ${\nabla }_T$ the tangential gradient on $\Gamma$, by ${\Delta }_T$ the tangential Laplacian on $\Gamma$ and by ${\partial }_{\nu }$ the normal derivative where $\nu$ represents the unit outward normal to $\Gamma$.

${\mu }_1,$ ${\mu }_2,$ ${\mu }_1^{\prime }$ and ${\mu }_2^{\prime }$ are positive real numbers, the two functions $g_1\left(u_t(t-\tau )\right)$ and $g_2\left(v_t(t-\tau )\right)$ describe the delays on the non-linear frictional dissipations $g_1\left(u_t\right)$ and $g_2\left(v_t\right)$, on $\Omega$ and ${\Gamma }_1$, respectively. $\tau >0$ is a time delay and $u_0,$ $v_0,$ $u_1,$ $v_{1,}$ $f_{0_1}$ and $f_{0_2}$ are the initial data in some suitable (Sobolev) function space.

Throughout history, the wave equation has known a great deal of research.

In our work, we are particularly interested in the wave equation with Wentzell-type boundary conditions, characterized by the presence of differential operators (${\Delta }_{T}u$) of the same order as the main operator.

These problems are involved in the modelling of many phenomena: mechanical-like elasticity, diffusion processes or wave propagation physics.

Wentzell-type conditions are obtained by asymptotic methods from transmission problems, (see Lemrabet K. [1]).

The following condition:

$${\partial }_{\nu }u-{\Delta }_{T}u=g,\mathrm{on}\Gamma$$

for this equation

$$-▵u+u=f,\mathrm{in}\Omega$$

was first introduced by Wentzell (Ventcel) in 1959, (see [2]), for diffusion processes. It models the heat exchange of the body $\Omega$ with the surrounding environment in the presence of a thin film, a very good conductor, on the surface of the body.

Delay is the property of a physical system by which the response to an applied force is slowed in its effect. Whenever material, information, or energy is physically transmitted from one place to another, there is a delay present in the law of feedback modelling the mechanical shift over time.

Delays often occur in many ways: physical problems, chemical, biological and economic phenomena.

The system (1) describes the vibrations of a flexible body with a thin boundary layer of high rigidity on its boundary ${\Gamma }_1$.

Our goal is to show that this problem is well posed, and that a unique solution exists.

1.1. Assumptions on the Damping and Delay Functions $g_i$ for $i=$ 1, 2

We pose the following assumptions on the damping and delay functions:

$\left(\mathbf{A}\mathbf{1}\right)$ $g_i$: $\mathbb{R}⟶\mathbb{R}$ is an odd non-decreasing function of the class ${\mathcal{C}}^1\left(\mathbb{R}\right)$, such that there exist r (sufficiently small), $c_i,$ $C_i,$ $c,$ ${\alpha }_1$ and ${\alpha }_2>0$ for $i=1,2,$ and a convex, increasing function H: ${\mathbb{R}}_{+}⟶{\mathbb{R}}_{+}$ of the class ${\mathcal{C}}^1\left({\mathbb{R}}_{+}\right)\cap {\mathcal{C}}^2\left(\right]0,\infty \left[\right)$ that satisfies:

$H\left(0\right)=0$ and H is linear on $[0,r]$ or $(H^{\prime }\left(0\right)=0$ and $H^{″}>0$ on $]0,r]),$ such that

$$c_i\left|s\right|\le \left|g_i\left(s\right)\right|\le C_i\left|s\right|\mathrm{if}\left|s\right|\ge r,$$

(2)

$$s^2+g_i^2\left(s\right)\le H^{-1}\left(s{g}_i\left(s\right)\right)\mathrm{if}\left|s\right|\le r,$$

(3)

$$\left|g_i^{\prime }\left(s\right)\right|\le c,$$

(4)

$${\alpha }_{1}s{g}_i\left(s\right)\le G_i\left(s\right)\le {\alpha }_{2}s{g}_i\left(s\right),$$

(5)

where

$$G_i\left(s\right)={\int }_0^s{g}_i\left(y\right)dy.$$

$\left(\mathbf{A}\mathbf{2}\right)$ ${\alpha }_2{\mu }_2<{\alpha }_1{\mu }_1$ and ${\alpha }_2{\mu }_2^{\prime }<{\alpha }_1{\mu }_1^{\prime }.$

1.2. Transformation of Problem (1)

Now, as in [3], we introduce the new variables:

$$\left\{\begin{array}{c}{z}_1(x,\rho ,t)=u_t(x,t-\rho \tau ),x\in \Omega ,\rho \in (0,1),t>0,\\ z_2(x,\rho ,t)=v_t(x,t-\rho \tau ),x\in {\Gamma }_1,\rho \in (0,1),t>0,\end{array}\right.$$

where $\tau >0$ is a time delay.

Then, we have

$$\left\{\begin{array}{c}\tau {\left(z_1\right)}_t(x,\rho ,t)+{\left(z_1\right)}_{\rho }(x,\rho ,t)=0,\mathrm{on}\Omega \times (0,1)\times (0,\infty ),\\ \tau {\left(z_2\right)}_t(x,\rho ,t)+{\left(z_2\right)}_{\rho }(x,\rho ,t)=0,\mathrm{on}{\Gamma }_1\times (0,1)\times (0,\infty ),\end{array}\right.$$

where ${\left(z_i\right)}_t=\frac{\partial z_i}{\partial t}$ and ${\left(z_i\right)}_{\rho }=\frac{\partial z_i}{\partial \rho }$ for $i=1,2$.

Therefore, problem (1) is equivalent to

$$\left\{\begin{array}{ccc}{u}_{tt}-\Delta u+{\mu }_1{g}_1\left(u_t\right)+{\mu }_2{g}_1\left(z_1(x,1,t)\right)=0,& \mathrm{in}& \Omega \times (0,\infty ),\\ v_{tt}+{\partial }_{\nu }u-{\Delta }_{T}v+{\mu }_1^{\prime }{g}_2\left(v_t\right)+{\mu }_2^{\prime }{g}_2\left(z_2(x,1,t)\right)=0,& \mathrm{on}& {\Gamma }_1\times (0,\infty ),\\ \tau {\left(z_1\right)}_t(x,\rho ,t)+{\left(z_1\right)}_{\rho }(x,\rho ,t)=0,& \mathrm{in}& \Omega \times (0,1)\times (0,\infty ),\\ \tau {\left(z_2\right)}_t(x,\rho ,t)+{\left(z_2\right)}_{\rho }(x,\rho ,t)=0,& \mathrm{on}& {\Gamma }_1\times (0,1)\times (0,\infty ),\\ u=v,& \mathrm{on}& \Gamma \times (0,\infty ),\\ u=0,& \mathrm{on}& {\Gamma }_0\times (0,\infty ),\\ z_1(x,0,t)=u_t(x,t),& \mathrm{in}& \Omega \times (0,\infty ),\\ z_2(x,0,t)=v_t(x,t),& \mathrm{on}& {\Gamma }_1\times (0,\infty ),\\ (u\left(0\right),v\left(0\right))=(u_0,v_0),& \mathrm{in}& \Omega \times \Gamma ,\\ (u_t\left(0\right),v_t\left(0\right))=(u_1,v_1),& \mathrm{in}& \Omega \times \Gamma ,\\ z_1(x,\rho ,0)=f_{0_1}\left(x,-\rho \tau \right),& \mathrm{in}& \Omega \times (0,1),\\ z_2(x,\rho ,0)=f_{0_2}\left(x,-\rho \tau \right),& \mathrm{on}& {\Gamma }_1\times (0,1).\end{array}\right.$$

(6)

1.3. Energy of System (6)

Let $\xi$ and $\zeta$ be strictly positive constants, such that

$$\tau \frac{{\mu }_2(1-{\alpha }_1)}{{\alpha }_1}<\xi <\tau \frac{{\mu }_1-{\alpha }_2{\mu }_2}{{\alpha }_2},$$

(7)

$$\tau \frac{{\mu }_2^{\prime }(1-{\alpha }_1)}{{\alpha }_1}<\zeta <\tau \frac{{\mu }_1^{\prime }-{\alpha }_2{\mu }_2^{\prime }}{{\alpha }_2}.$$

(8)

We define the energy associated with the solution to problem (6) by

$$\begin{array}{cc}\hfill E\left(t\right)& =\frac{1}{2}{∥u_t∥}^2+\frac{1}{2}{∥\nabla u∥}^2+\frac{1}{2}{∥v_t∥}_{{\Gamma }_1}^2+\frac{1}{2}{∥{\nabla }_{T}v∥}_{{\Gamma }_1}^2\hfill \\ \hfill & +\xi {\int }_{\Omega }\left({\int }_0^1{G}_1\left(z_1(x,\rho ,t)\right)d\rho \right)dx+\zeta {\int }_{{\Gamma }_1}\left({\int }_0^1{G}_2\left(z_2(x,\rho ,t)\right)d\rho \right)d\sigma ,\hfill \end{array}$$

(9)

where $∥.∥={\left(.,.\right)}^{\frac{1}{2}}$ and ${∥.∥}_{{\Gamma }_1}={\left(.,.\right)}_{{\Gamma }_1}^{\frac{1}{2}},$ (the norms associated with the inner products in $L^2\left(\Omega \right)$ and $L^2\left({\Gamma }_1\right)$, respectively).

1.4. Energy Decay

Therefore, we have the following lemma on the dissipation of energy $E\left(t\right)$:

Lemma 1.

Let $(u,v,z_1,z_2)$ be a solution to problem (6). Then, the energy functional defined by (9) satisfies

$$\begin{array}{ccc}\hfill E^{\prime }\left(t\right)& \le & -a_1{\int }_{\Omega }{u}_t{g}_1\left(u_t\right)dx-a_2{\int }_{{\Gamma }_1}{v}_t{g}_2\left(v_t\right)d\sigma \hfill \\ & & -a_3{\int }_{\Omega }{z}_1(x,1,t))g_1\left(z_1(x,1,t)\right)dx\hfill \\ & & -a_4{\int }_{{\Gamma }_1}{z}_2(x,1,t))g_2\left(z_2(x,1,t)\right)d\sigma \hfill \\ & \le & 0,\forall t\ge 0,\hfill \end{array}$$

(10)

where $a_1=\left({\mu }_1-\frac{\xi }{\tau }{\alpha }_2-{\mu }_2{\alpha }_2\right),$ $a_2=\left({\mu }_1^{\prime }-\frac{\zeta }{\tau }{\alpha }_2-{\mu }_2^{\prime }{\alpha }_2\right),$ $a_3=\left({\alpha }_1\frac{\xi }{\tau }-{\mu }_2(1-{\alpha }_1)\right)$ and $a_4=\left({\alpha }_1\frac{\zeta }{\tau }-{\mu }_2^{\prime }(1-{\alpha }_1)\right).$

For the proof of Lemma 1, see [4].

2. The Main Results

We introduce the following set

$$H_{{\Gamma }_0}^1\left(\Omega \right)=\left\{u\in H^1(\Omega )/{\left.u\right|}_{{\Gamma }_0}=0\right\},$$

which is given the Hilbert structure induced by $H^1\left(\Omega \right)$.

Then, we consider the canonical norms of $H_{{\Gamma }_0}^1\left(\Omega \right)$ and $H^1\left({\Gamma }_1\right)$

$${∥u∥}_{H_{{\Gamma }_0}^1\left(\Omega \right)}^2={∥\nabla u∥}^2,{∥v∥}_{H^1\left({\Gamma }_1\right)}^2={∥{\nabla }_{T}v∥}_{{\Gamma }_1}^2.$$

Now, we state the following unique result exists:

Theorem 1.

Let $(u_0,u_1,v_0,v_1)\in [H^2(\Omega )\cap H_{{\Gamma }_0}^1\left(\Omega \right))\times H_{{\Gamma }_0}^1\left(\Omega \right)]\times [H^2\left({\Gamma }_1\right)\times H^1\left({\Gamma }_1\right)],$ $f_{0_1}\in H_{{\Gamma }_0}^1\left(\Omega ;H^1\left(0,1\right)\right)$ and $f_{0_2}\in H^1\left({\Gamma }_1;H^1\left(0,1\right)\right)$ satisfying the following compatibility condition:

$$\left\{\begin{array}{c}{\partial }_{\nu }{u}_0-{\Delta }_T{v}_0+{\mu }_1^{\prime }{g}_2\left(v_1\right)=0,on{\Gamma }_1,\\ f_{0_1}(.,0)=u_t,in\Omega ,\\ f_{0_2}(.,0)=v_t,on{\Gamma }_1.\end{array}\right.$$

(11)

Assume that $\left(\mathbf{A}\mathbf{1}\right)$ and $\left(\mathbf{A}\mathbf{2}\right)$ hold, then problem (6) possesses a unique globally weak solution verifying $T>0$:

$$\begin{array}{cc}\hfill (u,u_t,u_{tt})& \in L^{\infty }(0,T;{\left[H_{{\Gamma }_0}^1\left(\Omega \right)\right]}^2\times L^2(\Omega )),\hfill \\ \hfill (v,v_t,v_{tt})& \in L^{\infty }(0,T;{\left[H^1\left({\Gamma }_1\right)\right]}^2\times L^2\left({\Gamma }_1\right)).\hfill \end{array}$$

The proof of Theorem 1 is given below using the Faedo-Galerkin approximation.

Proof of Theorem 1.

Throughout this proof, assume $(u_0,v_0)\in (H^2(\Omega )\cap H_{{\Gamma }_0}^1\left(\Omega \right))\times \left(H^2\left({\Gamma }_1\right)\cap H^1\left({\Gamma }_1\right)\right),$ $(u_1,v_1)\in H_{{\Gamma }_0}^1\left(\Omega \right)\times H^1\left({\Gamma }_1\right),$ $f_{0_1}\in H_{{\Gamma }_0}^1\left(\Omega ;H^1\left(0,1\right)\right)$ and $f_{0_2}\in H^1$ $\left({\Gamma }_1;H^1\left(0,1\right)\right).$

For any $n\in \mathbb{N}$, we denote by $U_n$ and $V_n$ the two finite dimensional spaces defined by, respectively, $U_n$ $=span\left\{w_1,w_2,...,w_n\right\}$ and $V_n=span\left\{{\tilde{w}}_1,{\tilde{w}}_2,...,{\tilde{w}}_n\right\},$ where ${\left\{w_i\right\}}_{1\le i\le n}$ and ${\left\{{\tilde{w}}_i\right\}}_{1\le i\le n}$ are the basis in the spaces $H^2(\Omega )\cap H_{{\Gamma }_0}^1\left(\Omega \right)$ and $H^2\left({\Gamma }_1\right)\cap H^1\left({\Gamma }_1\right)$, respectively.

Now, define for $1\le i\le n$ the sequences ${\varphi }_i(x,\rho )$ and ${\tilde{\varphi }}_i(x,\rho )$ as follows:

$$\left\{\begin{array}{c}{\varphi }_i(x,0)=w_i,\\ {\tilde{\varphi }}_i(x,0)={\tilde{w}}_i,\end{array}\right.$$

then extend ${\varphi }_i(x,0)$ by ${\varphi }_i(x,\rho )$ over $L^2\left(\Omega \times (0,1)\right)$, and ${\tilde{\varphi }}_i(x,0)$ by ${\tilde{\varphi }}_i(x,\rho )$ over $L^2\left({\Gamma }_1\times (0,1)\right)$, and denote $Z_n,$ ${\tilde{Z}}_n$ as the linear spaces generated by $\left\{{\varphi }_1,{\varphi }_2,...,{\varphi }_n\right\}$ and $\left\{{\tilde{\varphi }}_1,{\tilde{\varphi }}_2,...,{\tilde{\varphi }}_n\right\},$ respectively.

Let us define the approximations $u^n,$ $v^n,$ $z_1^n$ and $z_2^n$ by

$$u^n\left(t\right)=\sum _{i=1}^n{a}_i^n\left(t\right)w_i,v^n\left(t\right)=\sum _{i=1}^n{b}_i^n\left(t\right){\tilde{w}}_i,z_1^n\left(t\right)=\sum _{i=1}^n{c}_i^n\left(t\right){\varphi }_i,z_2^n\left(t\right)=\sum _{i=1}^n{d}_i^n\left(t\right){\tilde{\varphi }}_i,$$

where $a_i^n,$ $b_i^n,$ $c_i^n$ and $d_i^n$ are from the class ${\mathcal{C}}^2$ and determined by the following differential equations:

$$\begin{array}{c}(u_{tt}^n,w_i)+(\nabla u^n,\nabla w_i)+{\mu }_1(g_1\left(u_t^n\right),w_i)+{\mu }_2(g_1\left(z_1^n(x,1,t)\right),w_i)\\ +{(v_{tt}^n,{\tilde{w}}_i)}_{{\Gamma }_1}+{({\nabla }_T{v}^n,{\nabla }_T{\tilde{w}}_i)}_{{\Gamma }_1}+{\mu }_1^{\prime }{(g_2\left(v_t^n\right),{\tilde{w}}_i)}_{{\Gamma }_1}\\ +{\mu }_2^{\prime }{(g_2\left(z_2^n(x,1,t)\right),{\tilde{w}}_i)}_{{\Gamma }_1}=0,1\le i\le n,\end{array}$$

(12)

$${\int }_{\Omega }{\int }_0^1(\tau z_{1_t}^n+z_{1_{\rho }}^n){\varphi }_{i}d\rho dx=0,1\le i\le n$$

(13)

and

$${\int }_{{\Gamma }_1}{\int }_0^1(\tau z_{2_t}^n+z_{2_{\rho }}^n){\tilde{\varphi }}_{i}d\rho d\sigma =0,1\le i\le n,$$

(14)

with initial data:

$$\left\{\begin{array}{ccccccccc}{u}^n\left(0\right)& =& u_0^n& =& \sum _{i=1}^n{a}_i^n\left(0\right)w_i& \to & u_0& \mathrm{in}& \left(H^2\left(\Omega \right)\cap H_{{\Gamma }_0}^1\left(\Omega \right)\right),\\ u_t^n\left(0\right)& =& u_1^n& =& \sum _{i=1}^n{\left(a_i^n\right)}_t\left(0\right)w_i& \to & u_1& \mathrm{in}& H_{{\Gamma }_0}^1\left(\Omega \right),\\ v^n\left(0\right)& =& v_0^n& =& \sum _{i=1}^n{b}_i^n\left(0\right){\tilde{w}}_i& \to & v_0& \mathrm{in}& \left(H^2\left({\Gamma }_1\right)\cap H^1\left({\Gamma }_1\right)\right),\\ v_t^n\left(0\right)& =& v_1^n& =& \sum _{i=1}^n{\left(b_i^n\right)}_t\left(0\right){\tilde{w}}_i& \to & v_1& \mathrm{in}& H^1\left({\Gamma }_1\right),\\ z_1^n(\rho ,0)& =& z_{0_1}^n& =& \sum _{i=1}^n{c}_i^n\left(0\right){\varphi }_i& \to & f_{0_1}& \mathrm{in}& H_{{\Gamma }_0}^1\left(\Omega ;H^1\left(0,1\right)\right),\\ z_2^n(\rho ,0)& =& z_{0_2}^n& =& \sum _{i=1}^n{d}_i^n\left(0\right){\tilde{\varphi }}_i& \to & f_{0_2}& \mathrm{in}& H^1\left({\Gamma }_1;H^1\left(0,1\right)\right).\end{array}\right.$$

(15)

The local existence of solutions to problems (12)–(15) is standard by the theory of ordinary differential equations. We can conclude that $t_n$ $>0$ such that in $[0,t_n]$, problems (12)–(15) have a unique local solution which can be extended to a maximum interval $[0,T]$ (with $0<T\le \infty$) by Zorn’s lemma, since the non-linear terms in (12) are locally Lipschitz continuous.

We can utilize a standard compactness argument for the limiting procedure and it suffices to derive some a priori estimates for $(u^n,v^n,z_1^n,z_2^n).$

The first estimate

Since the sequences ${\left(u_0^n\right)}_n,$ ${\left(u_1^n\right)}_n,$ ${\left(v_0^n\right)}_n,$ ${\left(v_1^n\right)}_n,$ ${\left(z_{0_1}^n\right)}_n$ and ${\left(z_{0_2}^n\right)}_n$ converge, the standard calculations, using (12)–(15) similar to those used to find (10), yield a number $M_1$ independent of n such that

$$\begin{array}{ccc}& & E_n\left(t\right)+a_1{\int }_0^t{\int }_{\Omega }{u}_t^n{g}_1\left(u_t^n\right)dxds+a_3{\int }_0^t{\int }_{\Omega }{z}_1^n(x,1,s)g_1\left(z_1^n(x,1,s)\right)dxds\hfill \\ & & +a_2{\int }_0^t{\int }_{{\Gamma }_1}{v}_t^n{g}_2\left(v_t^n\right)d\sigma ds+a_4{\int }_0^t{\int }_{{\Gamma }_1}{z}_2^n(x,1,s)g_2\left(z_2^n(x,1,s)\right)d\sigma ds\hfill \\ & \le & E_n\left(0\right)\le M_1,\hfill \end{array}$$

(16)

where

$$\begin{array}{ccc}\hfill E_n\left(t\right)& =& \frac{1}{2}\left({∥u_t^n∥}^2+{∥\nabla u^n∥}^2+{∥v_t^n∥}_{{\Gamma }_1}^2+{∥{\nabla }_T{v}^n∥}_{{\Gamma }_1}^2\right)\hfill \\ & & +\xi {\int }_{\Omega }\left({\int }_0^1{G}_1(z_1^n(x,\rho ,t)d\rho \right)dx+\zeta {\int }_{{\Gamma }_1}\left({\int }_0^1{G}_2(z_2^n(x,\rho ,t)d\rho \right)d\sigma \hfill \end{array}$$

and $a_i,$ $i=1,...,4$ are defined in Lemma 1.

The estimate (16) implies that the solution ${(u^n,v^n,z_1^n,z_2^n)}_n$ exists globally in $[0,+\infty ).$

Estimate (16) yields for any $T>0$

$$u^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;H_{{\Gamma }_0}^1(\Omega )\right),$$

(17)

$$v^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;H^1\left({\Gamma }_1\right)\right),$$

(18)

$$u_t^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;L^2(\Omega )\right),$$

(19)

$$v_t^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;L^2\left({\Gamma }_1\right)\right),$$

(20)

$$u_t^n{g}_1\left(u_t^n\right)\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^1(\Omega \times (0,T)),$$

(21)

$$v_t^n{g}_2\left(v_t^n\right)\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^1({\Gamma }_1\times (0,T)),$$

(22)

$$G_1\left(z_1^n\right))\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;L^1(\Omega \times (0,1))\right),$$

(23)

$$G_2\left(z_2^n\right))\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;L^1({\Gamma }_1\times (0,1))\right),$$

(24)

$$z_1^n(x,1,t)g_1\left(z_1^n(x,1,t)\right)\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^1(\Omega \times (0,T)),$$

(25)

$$z_2^n(x,1,t)g_2\left(z_2^n(x,1,t)\right)\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^1({\Gamma }_1\times (0,T)).$$

(26)

The second estimate

We need to estimate $u_{tt}^n\left(0\right)$ and $v_{tt}^n\left(0\right)$ in norms $L^2(\Omega )$ and $L^2\left({\Gamma }_1\right)$, respectively. By taking $t=0$ and considering $w_i=u_{tt}^n\left(0\right)$ and ${\tilde{w}}_i=v_{tt}^n\left(0\right)$ in (12), we obtain

$$\begin{array}{ccc}& & {∥u_{tt}^n\left(0\right)∥}^2+(\nabla u_0^n,\nabla u_{tt}^n\left(0\right))+{\mu }_1(g_1\left(u_1^n\right),u_{tt}^n\left(0\right))+{\mu }_2(g_1\left(z_{0_1}^n\right),u_{tt}^n\left(0\right))\hfill \\ & & +{∥v_{tt}^n\left(0\right)∥}_{{\Gamma }_1}^2+{({\nabla }_T{v}_0^n,{\nabla }_T{v}_{tt}^n\left(0\right))}_{{\Gamma }_1}+{\mu }_1^{\prime }{(g_2\left(v_1^n\right),v_{tt}^n\left(0\right))}_{{\Gamma }_1}+{\mu }_2^{\prime }{(g_2\left(z_{0_2}^n\right),v_{tt}^n\left(0\right))}_{{\Gamma }_1}\hfill \\ & =& 0.\hfill \end{array}$$

(27)

We have the equalities

$$(\nabla u_0^n,\nabla u_{tt}^n\left(0\right))=-(\Delta u_0^n,u_{tt}^n\left(0\right))+{({\partial }_{\nu }{u}_0^n,v_{tt}^n\left(0\right))}_{{\Gamma }_1},$$

(28)

$${({\nabla }_T{v}_0^n,{\nabla }_T{v}_{tt}^n\left(0\right))}_{{\Gamma }_1}=-{({\Delta }_T{v}_0^n,v_{tt}^n\left(0\right))}_{{\Gamma }_1}.$$

(29)

Employing Young’s inequality on (28) and (29) and using the fact that if $u_0^n\in \left(H_{{\Gamma }_0}^1\left(\Omega \right)\right.$ $\left.\cap H^2\left(\Omega \right)\right),$ then ${\partial }_{\nu }{u}_0^n\in H^{1/2}\left({\Gamma }_1\right)\hookrightarrow L^2\left({\Gamma }_1\right);$ hence, ${\partial }_{\nu }{u}_0^n\in L^2\left({\Gamma }_1\right)$; we obtain,

$$\begin{array}{ccc}\hfill (\nabla u_0^n,\nabla u_{tt}^n\left(0\right))& =& -(\Delta u_0^n,u_{tt}^n\left(0\right))+{({\partial }_{\nu }{u}_0^n,v_{tt}^n\left(0\right))}_{{\Gamma }_1}\hfill \\ & \le & \frac{1}{4\epsilon }{∥\Delta u_0^n∥}^2+\frac{1}{4\epsilon }{∥{\partial }_{\nu }{u}_0^n∥}_{{\Gamma }_1}^2+\epsilon {∥u_{tt}^n\left(0\right)∥}^2+\epsilon {∥v_{tt}^n(0∥}_{{\Gamma }_1}^2,\hfill \end{array}$$

(30)

$$-{({\Delta }_T{v}_0^n,v_{tt}^n\left(0\right))}_{{\Gamma }_1}\le \frac{1}{4\epsilon }{∥{\Delta }_T{v}_0^n∥}_{{\Gamma }_1}^2+\epsilon {∥v_{tt}^n\left(0\right)∥}_{{\Gamma }_1}^2,$$

(31)

$${\mu }_1(g_1\left(u_1^n\right),u_{tt}^n\left(0\right))\le \frac{{\mu }_1^2}{4\epsilon }{∥g_1\left(u_1^n\right)∥}^2+\epsilon {∥u_{tt}^n\left(0\right)∥}^2,$$

(32)

$${\mu }_1^{\prime }{(g_2\left(v_1^n\right),v_{tt}^n\left(0\right))}_{{\Gamma }_1}\le \frac{{\left({\mu }_1^{\prime }\right)}^2}{4\epsilon }{∥g_2\left(v_1^n\right)∥}_{{\Gamma }_1}^2+\epsilon {∥v_{tt}^n\left(0\right)∥}_{{\Gamma }_1}^2,$$

(33)

$${\mu }_2(g_1\left(z_{0_1}^n\right),u_{tt}^n\left(0\right))\le \frac{{\mu }_2^2}{4\epsilon }{∥g_1\left(z_{0_1}^n\right)∥}^2+\epsilon {∥u_{tt}^n\left(0\right)∥}^2,$$

(34)

$${\mu }_2^{\prime }{(g_2\left(z_{0_2}^n\right),v_{tt}^n\left(0\right))}_{{\Gamma }_1}\le \frac{{\left({\mu }_2^{\prime }\right)}^2}{4\epsilon }{∥g_2\left(z_{0_2}^n\right)∥}_{{\Gamma }_1}^2+\epsilon {∥v_{tt}^n\left(0\right)∥}_{{\Gamma }_1}^2,$$

(35)

by re-introducing (30)–(35) in (27), with a suitable choice of $\epsilon$ and since ${\left(g_1\left(u_1^n\right)\right)}_n,$ ${\left(g_1\left(z_{0_1}^n\right)\right)}_n$ and ${\left(g_2\left(v_1^n\right)\right)}_n,$ ${\left(g_2\left(z_{0_2}^n\right)\right)}_n$ are bounded in $L^2(\Omega )$ and $L^2\left({\Gamma }_1\right)$, respectively, by $\left(\mathbf{A}\mathbf{1}\right),$ $\left(\mathbf{A}\mathbf{2}\right)$ and initial data (15), we obtain

$$∥u_{tt}^n\left(0\right)∥+{∥v_{tt}^n\left(0\right)∥}_{{\Gamma }_1}\le M_2,$$

(36)

where $M_2$ is a positive constant independent of n and depends on the initial data.

Next, differentiating (12) with respect to t, multiplying the resulting equation by ${\left(a_i^n\right)}_{tt}\left(t\right)$ in $\Omega$ and ${\left(b_i^n\right)}_{tt}\left(t\right)$ on ${\Gamma }_1$, and summing over i from 1 to n, we obtain

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}\left\{{∥u_{tt}^n∥}^2+{∥\nabla u_t^n∥}^2+{∥v_{tt}^n∥}_{{\Gamma }_1}^2+{∥{\nabla }_T{v}_t^n∥}_{{\Gamma }_1}^2\right\}\hfill \\ & & +{\mu }_1{\int }_{\Omega }{\left|u_{tt}^n\right|}^2{\left(g_1\right)}_t\left(u_t^n\right)dx+{\mu }_2{\int }_{\Omega }{u}_{tt}^n{\left(z_1^n\right)}_t(x,1,t){\left(g_1\right)}_t\left(z_1^n(x,1,t)\right)dx\hfill \\ & & +{\mu }_1^{\prime }{\int }_{{\Gamma }_1}{\left|v_{tt}^n\right|}^2{\left(g_2\right)}_t\left(v_t^n\right)d\sigma +{\mu }_2^{\prime }{\int }_{{\Gamma }_1}{v}_{tt}^n{\left(z_2^n\right)}_t(x,1,t){\left(g_2\right)}_t\left(z_2^n(x,1,t)\right)d\sigma \hfill \\ & =& 0.\hfill \end{array}$$

(37)

Differentiating (13) with respect to t, multiplying the resulting equation by ${\left(c_i^n\right)}_t\left(t\right)$ and summing over i from 1 to n, it follows that

$$\frac{\tau }{2}\frac{d}{dt}{∥z_{1_t}^n\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2+\frac{1}{2}\frac{d}{d\rho }{∥z_{1_t}^n\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2=0.$$

(38)

Analogously, differentiating (14) with respect to t, multiplying the resulting equation by ${\left(d_i^n\right)}_t\left(t\right)$ and summing over i from 1 to n, it follows that

$$\frac{\tau }{2}\frac{d}{dt}{∥z_{2_t}^n\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2+\frac{1}{2}\frac{d}{d\rho }{∥z_{2_t}^n\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2=0.$$

(39)

Taking the sum of (37)–(39), we obtain

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}\left\{{∥u_{tt}^n∥}^2+{∥\nabla u_t^n∥}^2+{∥v_{tt}^n∥}_{{\Gamma }_1}^2+{∥{\nabla }_T{v}_t^n∥}_{{\Gamma }_1}^2\right.\hfill \\ & & \left.+\tau {∥{\left(z_1^n\right)}_t\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2+\tau {∥{\left(z_2^n\right)}_t\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2\right\}\hfill \\ & & +{\mu }_1{\int }_{\Omega }{\left|u_{tt}^n\right|}^2{\left(g_1\right)}_t\left(u_t^n\right)dx+\frac{1}{2}{\int }_{\Omega }{\left|{\left(z_1^n\right)}_t(x,1,t)\right|}^{2}dx\hfill \\ & & +{\mu }_1^{\prime }{\int }_{{\Gamma }_1}{\left|v_{tt}^n\right|}^2{\left(g_2\right)}_t\left(v_t^n\right)d\sigma +\frac{1}{2}{\int }_{{\Gamma }_1}{\left|{\left(z_2^n\right)}_t(x,1,t)\right|}^{2}d\sigma \hfill \\ & =& \frac{1}{2}{∥u_{tt}^n∥}^2-{\mu }_2{\int }_{\Omega }{u}_{tt}^n{\left(z_1^n\right)}_t(x,1,t){\left(g_1\right)}_t\left(z_1^n(x,1,t)\right)dx\hfill \\ & & +\frac{1}{2}{∥v_{tt}^n∥}_{{\Gamma }_1}^2-{\mu }_2^{\prime }{\int }_{{\Gamma }_1}{v}_{tt}^n{\left(z_2^n\right)}_t(x,1,t){\left(g_2\right)}_t\left(z_2^n(x,1,t)\right)d\sigma .\hfill \end{array}$$

(40)

Using (4) and Young’s inequality, we obtain

$$\begin{array}{ccc}& & {\mu }_2{\int }_{\Omega }{u}_{tt}^n{\left(z_1^n\right)}_t(x,1,t){\left(g_1\right)}_t\left(z_1^n(x,1,t)\right)dx\hfill \\ & \le & \epsilon {\int }_{\Omega }{\left|{\left(z_1^n\right)}_t(x,1,t)\right|}^{2}dx+\frac{{\left({\mu }_{2}c\right)}^2}{4\epsilon }{∥u_{tt}^n∥}^2,\hfill \end{array}$$

(41)

$$\begin{array}{ccc}& & {\mu }_2^{\prime }{\int }_{{\Gamma }_1}{v}_{tt}^n{\left(z_2^n\right)}_t(x,1,t){\left(g_2\right)}_t\left(z_2^n(x,1,t)\right)d\sigma \hfill \\ & \le & \epsilon {\int }_{{\Gamma }_1}{\left|{\left(z_2^n\right)}_t(x,1,t)\right|}^{2}d\sigma +\frac{{\left({\mu }_2^{\prime }c\right)}^2}{4\epsilon }{∥v_{tt}^n∥}_{{\Gamma }_1}^2.\hfill \end{array}$$

(42)

Re-introducing (41) and (42) into (40), and choosing $\epsilon$ small enough, we obtain

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}\left\{{∥u_{tt}^n∥}^2+{∥\nabla u_t^n∥}^2+{∥v_{tt}^n∥}_{{\Gamma }_1}^2+{∥{\nabla }_T{v}_t^n∥}_{{\Gamma }_1}^2\right.\hfill \\ & & \left.+\tau {∥{\left(z_1^n\right)}_t\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2+\tau {∥{\left(z_2^n\right)}_t\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2\right\}\hfill \\ & & +{\mu }_1{\int }_{\Omega }{\left|u_{tt}^n\right|}^2{\left(g_1\right)}_t\left(u_t^n\right)dx+c{\int }_{\Omega }{\left|{\left(z_1^n\right)}_t(x,1,t)\right|}^{2}dx\hfill \\ & & +{\mu }_1^{\prime }{\int }_{{\Gamma }_1}{\left|v_{tt}^n\right|}^2{\left(g_2\right)}_t\left(v_t^n\right)d\sigma +c{\int }_{{\Gamma }_1}{\left|{\left(z_2^n\right)}_t(x,1,t)\right|}^{2}d\sigma \hfill \\ & \le & c^{\prime }\left\{{∥u_{tt}^n∥}^2+{∥v_{tt}^n∥}_{{\Gamma }_1}^2\right\}.\hfill \end{array}$$

Integrating the last inequality over $(0,t)$, we obtain

$$\begin{array}{ccc}& & \frac{1}{2}\left\{{∥u_{tt}^n∥}^2+{∥\nabla u_t^n∥}^2+{∥v_{tt}^n∥}_{{\Gamma }_1}^2+{∥{\nabla }_T{v}_t^n∥}_{{\Gamma }_1}^2\right.\hfill \\ & & \left.+\tau {∥{\left(z_1^n\right)}_t\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2+\tau {∥{\left(z_2^n\right)}_t\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2\right\}\hfill \\ & & +{\mu }_1{\int }_0^t{\int }_{\Omega }{\left|u_{tt}^n\right|}^2{\left(g_1\right)}_t\left(u_t^n\right)dxds+c{\int }_0^t{\int }_{\Omega }{\left(z_1^n\right)}_t{\left|(x,1,t)\right|}^{2}dxds\hfill \\ & & +{\mu }_1^{\prime }{\int }_0^t{\int }_{{\Gamma }_1}{\left|v_{tt}^n\right|}^2{\left(g_2\right)}_t\left(v_t^n\right)d\sigma ds+c{\int }_0^t{\int }_{{\Gamma }_1}{\left|{\left(z_2^n\right)}_t(x,1,t)\right|}^{2}d\sigma ds\hfill \\ & \le & \frac{1}{2}\left\{{∥u_{tt}^n\left(0\right)∥}^2+{∥v_{tt}^n\left(0\right)∥}_{{\Gamma }_1}^2+{∥\nabla u_1^n∥}^2+{∥{\nabla }_T{v}_1^n∥}_{{\Gamma }_1}^2\right.\hfill \\ & & \left.+\tau {∥{\left(z_1^n\right)}_t\left(x,\rho ,0\right)∥}_{L^2\left(\Omega \times (0,1\right))}^2+\tau {∥{\left(z_2^n\right)}_t\left(x,\rho ,0\right)∥}_{L^2\left({\Gamma }_1\times (0,1\right))}^2\right\}\hfill \\ & & +c^{\prime }\left\{{\int }_0^t{∥u_{tt}^n\left(s\right)∥}_{\Omega }^{2}ds+{\int }_0^t{∥v_{tt}^n\left(s\right)∥}_{{\Gamma }_1}^{2}ds\right\}.\hfill \end{array}$$

Using (15) and (36), and then Gronwall’s lemma, to obtain

$$\begin{array}{ccc}& & {∥u_{tt}^n∥}^2+{∥\nabla u_t^n∥}^2+{∥v_{tt}^n∥}_{{\Gamma }_1}^2+{∥{\nabla }_T{v}_t^n∥}_{{\Gamma }_1}^2+\tau {∥{\left(z_1^n\right)}_t\left(\rho ,t\right)∥}_{L^2\left(\Omega \times (0,1\right))}^2\hfill \\ & & +\tau {∥{\left(z_2^n\right)}_t\left(\rho ,t\right)∥}_{L^2\left({\Gamma }_1\times (0,1\right))}^2+{\mu }_1{\int }_0^t{\int }_{\Omega }{\left|u_{tt}^n\right|}^2{\left(g_1\right)}_t\left(u_t^n\right)dxds\hfill \\ & & +c{\int }_0^t{\int }_{\Omega }{\left(z_1^n\right)}_t{\left|(x,1,s)\right|}^{2}dxds+{\mu }_1^{\prime }{\int }_0^t{\int }_{{\Gamma }_1}{\left|v_{tt}^n\right|}^2{\left(g_2\right)}_t\left(v_t^n\right)d\sigma ds\hfill \\ & & +c{\int }_0^t{\int }_{{\Gamma }_1}{\left|{\left(z_2^n\right)}_t(x,1,s)\right|}^{2}d\sigma ds\hfill \\ & \le & M_3,\hfill \end{array}$$

where $M_3$ is independent of n and for all $t\in \left[0,T\right]$. Therefore, we conclude that

$$u_t^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;H_{{\Gamma }_0}^1(\Omega )\right),$$

(43)

$$v_t^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;H^1\left({\Gamma }_1\right)\right),$$

(44)

$$u_{tt}^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;L^2(\Omega )\right),$$

(45)

$$v_{tt}^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;L^2\left({\Gamma }_1\right)\right),$$

(46)

$$z_{1_t}^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega \times (0,1)),$$

(47)

$$z_{2_t}^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;L^2({\Gamma }_1\times (0,1)).$$

(48)

Estimate for  ${\left(z_1^n\right)}_n$  and  ${\left(z_2^n\right)}_n$

Replacing ${\varphi }_i$ with $-\Delta {\varphi }_i$ in (13), multiplying the resulting equation by $c_i^n\left(t\right)$ and summing over i from 1 to n, it follows that

$$\frac{\tau }{2}\frac{d}{dt}{∥\nabla z_1^n\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2+\frac{1}{2}\frac{d}{d\rho }{∥\nabla z_1^n\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2=0.$$

(49)

Similarly, replacing ${\tilde{\varphi }}_i$ with $-{\Delta }_T{\tilde{\varphi }}_i$ in (14), multiplying the resulting equation by $d_i^n\left(t\right)$ and summing over i from 1 to n, it follows that

$$\frac{\tau }{2}\frac{d}{dt}{∥{\nabla }_T{z}_2^n\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2+\frac{1}{2}\frac{d}{d\rho }{∥{\nabla }_T{z}_2^n\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2=0.$$

(50)

Combining (49) and (50), we have

$$\begin{array}{ccc}& & \frac{\tau }{2}\frac{d}{dt}{∥\nabla z_1^n\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2+\frac{\tau }{2}\frac{d}{dt}{∥{\nabla }_T{z}_2^n\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2\hfill \\ & & +\frac{1}{2}\left\{{\int }_{\Omega }{\left|\nabla z_1^n(x,1,t)\right|}^{2}dx+{\int }_{{\Gamma }_1}{\left|{\nabla }_T{z}_2^n(x,1,t)\right|}^{2}d\sigma \right\}\hfill \\ & =& \frac{1}{2}\left\{{∥\nabla u^n∥}^2+{∥{\nabla }_T{v}^n∥}_{{\Gamma }_1}^2\right\}.\hfill \end{array}$$

Integrating the last inequality over $(0,t)$ and using Gronwall’s lemma, we have

$$\begin{array}{ccc}& & \frac{\tau }{2}{∥\nabla z_1^n\left(\rho ,t\right)∥}_{L^2(\Omega \times (0,1))}^2+\frac{\tau }{2}{∥{\nabla }_T{z}_2^n\left(\rho ,t\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2\hfill \\ & \le & e^{cT}\left\{\frac{\tau }{2}{∥\nabla z_1^n\left(x,\rho ,0\right)∥}_{L^2(\Omega \times (0,1))}^2+\frac{\tau }{2}{∥{\nabla }_T{z}_2^n\left(x,\rho ,0\right)∥}_{L^2({\Gamma }_1\times (0,1))}^2\right\},\hfill \end{array}$$

for all $t\in \left[0,T\right]$. Therefore, we conclude that

$$z_1^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;H_{{\Gamma }_0}^1\left(\Omega ;L^2\left(0,1\right)\right)\right),$$

(51)

$$z_2^n\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }\left(0,T;H^1\left({\Gamma }_1;L^2\left(0,1\right)\right)\right).$$

(52)

Passing the limit

Applying Dunford–Petti’s theorem, we conclude that there exists subsequences of ${\left(u^n\right)}_n,$ ${\left(v^n\right)}_n,$ ${\left(z_1^n\right)}_n$ and ${\left(z_2^n\right)}_n$ which we still denote by ${\left(u^n\right)}_n,$ ${\left(v^n\right)}_n,$ ${\left(z_1^n\right)}_n$ and ${\left(z_2^n\right)}_n$, respectively, such that from (17), (43) and (45), we obtain

$$(u^n,u_t^n,u_{tt}^n)\rightharpoonup (u,u_t,u_{tt})\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }\left(0,T;{\left[H_{{\Gamma }_0}^1(\Omega )\right]}^2\times L^2\left(\Omega \right)\right),$$

(53)

from (18), (44) and (46), we obtain

$$(v^n,v_t^n,v_{tt}^n)\rightharpoonup (v,v_t,v_{tt})\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }\left(0,T;{\left[H^1\left({\Gamma }_1\right)\right]}^2\times L^2\left({\Gamma }_1\right)\right),$$

(54)

from (51) and (52), we find

$$z_1^n\rightharpoonup z_1\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }\left(0,T;H_{{\Gamma }_0}^1\left(\Omega ;L^2\left(0,1\right)\right)\right),$$

(55)

$$z_2^n\rightharpoonup z_2\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }\left(0,T;H^1\left({\Gamma }_1;L^2\left(0,1\right)\right)\right),$$

(56)

from (47) and (48), we obtain

$$z_{1_t}^n\rightharpoonup z_{1_t}\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega \times (0,1))),$$

(57)

$$z_{2_t}^n\rightharpoonup z_{2_t}^n\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^{\infty }(0,T;L^2({\Gamma }_1\times (0,1))),$$

(58)

and from (19)–(26), we have

$$g_1\left(u_t^n\right)\rightharpoonup {\chi }_1\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^2((0,T)\times \Omega ),$$

$$g_2\left(v_t^n\right)\rightharpoonup {\chi }_2\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^2((0,T)\times {\Gamma }_1),$$

$$g_1\left(z_1^n(x,1,t)\right)\rightharpoonup {\Psi }_1\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^2((0,T)\times \Omega ),$$

$$g_2\left(z_2^n(x,1,t)\right)\rightharpoonup {\Psi }_2\mathrm{weakly}\mathrm{star}\mathrm{in}{L}^2((0,T)\times {\Gamma }_1).$$

Thanks to Aubin–Lions’s theorem, (see [5]), we deduce that there exists subsequences which we still denote by ${\left(u^n\right)}_n$, ${\left(v^n\right)}_n$, ${\left(z_1^n\right)}_n$ and ${\left(z_2^n\right)}_n,$ such that

$$u^n\to u\mathrm{strongly}\mathrm{in}{L}^2\left(0,T;L^2(\Omega )\right),$$

(59)

$$u_t^n\to u_t\mathrm{strongly}\mathrm{in}{L}^2\left(0,T;L^2(\Omega )\right),$$

(60)

$$v^n\to v\mathrm{strongly}\mathrm{in}{L}^2\left(0,T;L^2\left({\Gamma }_1\right)\right),$$

(61)

$$v_t^n\to v_t\mathrm{strongly}\mathrm{in}{L}^2\left(0,T;L^2\left({\Gamma }_1\right)\right),$$

(62)

$$z_1^n\to z_1\mathrm{strongly}\mathrm{in}{L}^2\left(\Omega \times \left(0,1\right)\times (0,T)\right),$$

(63)

$$z_2^n\to z_2\mathrm{strongly}\mathrm{in}{L}^2\left({\Gamma }_1\times \left(0,1\right)\times (0,T)\right).$$

(64)

Analysis of the non-linear terms

Denoted by $Q=\Omega \times (0,T)$ and $\Sigma ={\Gamma }_1\times (0,T)$.

We can deduce from (60) and (62) that

$$u_t^n\to u_t\mathrm{almost}\mathrm{everywhere}\mathrm{on}Q,$$

(65)

$$v_t^n\to v_t\mathrm{almost}\mathrm{everywhere}\mathrm{on}\Sigma ,$$

(66)

and

$$\begin{array}{ccc}\hfill z_1^n& \to & z_1\mathrm{strongly}\mathrm{in}{L}^2\left(0,T;L^2(\Omega )\right)\mathrm{and}a.e\mathrm{on}Q,\hfill \\ \hfill z_2^n& \to & z_2\mathrm{strongly}\mathrm{in}{L}^2\left(0,T;L^2\left({\Gamma }_1\right)\right)\mathrm{and}a.e\mathrm{on}\Sigma .\hfill \end{array}$$

We have the following two lemmas, (for the proof, see [4]):

Lemma 2.

For each $T>0,$ $g_1\left(u_t\right),$ $g_1\left(z_1(.,1,.)\right)\in L^1\left(Q\right)$ and $g_2\left(v_t\right),$ $g_2\left(z_2(.,1,.)\right)\in L^1(\Sigma ),$ we have

$${∥g_1\left(u_t\right)∥}_{L^1\left(Q\right)},{∥g_1\left(z_1(.,1,.)\right)∥}_{L^1\left(Q\right)}\le A_1$$

and

$${∥g_2\left(v_t\right)∥}_{L^1(\Sigma )},{∥g_2\left(z_2(.,1,.)\right)∥}_{L^1(\Sigma )}\le A_2,$$

where $A_1$ and $A_2$ are constants independent of t.

Lemma 3.

We have the following convergences

$$\left\{\begin{array}{c}{g}_1\left(u_t^n\right)\to g_1\left(u_t\right)\mathrm{in}{L}^1(\Omega \times (0,T)),\\ g_2\left(v_t^n\right)\to g_2\left(v_t\right)\mathrm{in}{L}^1({\Gamma }_1\times (0,T)).\end{array}\right.$$

$$\left\{\begin{array}{c}{g}_1\left(z_1^n\right)\to g_1\left(z_1\right)\mathrm{in}{L}^1(\Omega \times (0,T)),\\ g_2\left(z_2^n\right)\to g_2\left(z_2\right)\mathrm{in}{L}^1({\Gamma }_1\times (0,T)).\end{array}\right.$$

Hence, from Lemma 3, we deduce that

$$\left\{\begin{array}{c}{g}_1\left(u_t^n\right)\rightharpoonup {\chi }_1=g_1\left(u_t\right)\mathrm{weakly}\mathrm{in}{L}^2(\Omega \times (0,T)),\\ g_2\left(v_t^n\right)\rightharpoonup {\chi }_2=g_2\left(v_t\right)\mathrm{weakly}\mathrm{in}{L}^2({\Gamma }_1\times (0,T)),\end{array}\right.$$

(67)

and

$$\left\{\begin{array}{c}{g}_1\left(z_1^n(x,1,t)\right)\rightharpoonup {\Psi }_1=g_1\left(z_1(x,1,t)\right)\mathrm{weakly}\mathrm{in}{L}^2(\Omega \times (0,T)),\\ g_2\left(z_2^n(x,1,t)\right)\rightharpoonup {\Psi }_2=g_2\left(z_2(x,1,t)\right)\mathrm{weakly}\mathrm{in}{L}^2({\Gamma }_1\times (0,T)).\end{array}\right.$$

(68)

Now, returning to (12) and using standard arguments, we can show from the above estimates that

$$u_{tt}-▵u+{\mu }_1{g}_1\left(u_t\right)+{\mu }_2{g}_1\left(z_1(.,1,.)\right)=0,\mathrm{in}{\mathcal{D}}^{\prime }\left(\Omega \times (0,T)\right).$$

(69)

Since $u_{tt},$ $g_1\left(u_t\right)$ and $g_1\left(z_1(.,1,.)\right)\in L^2\left(0,T;L^2(\Omega )\right),$ we obtain from identity (69)

$$▵u\in L^2\left(0,T;L^2(\Omega )\right),$$

and therefore identity (69) yields

$$u_{tt}-▵u+{\mu }_1{g}_1\left(u_t\right)+{\mu }_2{g}_1\left(z_1(.,1,.)\right)=0,\mathrm{in}{L}^2\left(0,T;L^2(\Omega )\right).$$

(70)

Taking (70) into account and making use of the generalized Green’s formula, we deduce that

$${\partial }_{\nu }u-{\Delta }_{T}v=-{\mu }_1^{\prime }{g}_2\left(v_t\right)-{\mu }_2^{\prime }{g}_2\left(z_2(.,1,.)\right)-v_{tt},\mathrm{in}{\mathcal{D}}^{\prime }\left(0,T;H^{-\frac{1}{2}}\left({\Gamma }_1\right)\right),$$

and since $v_{tt},$ $g_2\left(v_t\right)$ and $g_2\left(z_2(.,1,.)\right)\in L^2\left(0,T;L^2\left({\Gamma }_1\right)\right),$ we infer that

$${\partial }_{\nu }u-{\Delta }_{T}v=-{\mu }_1^{\prime }{g}_2\left(v_t\right)-{\mu }_2^{\prime }{g}_2\left(z_2(.,1,.)\right)-v_{tt},\mathrm{in}{L}^2\left(0,T;L^2\left({\Gamma }_1\right)\right),$$

then

$$v_{tt}+{\partial }_{\nu }u-{\Delta }_{T}v+{\mu }_1^{\prime }{g}_2\left(v_t\right)+{\mu }_2^{\prime }{g}_2\left(z_2(.,1,.)\right)=0,\mathrm{in}{L}^2\left(0,T;L^2\left({\Gamma }_1\right)\right).$$

Exploiting convergences (55)–(58), (63) and (64), we pass to the limit in (13) and (14) to obtain

$${\int }_0^T{\int }_0^1{\int }_{\Omega }(\tau z_{1_t}+z_{1\rho }){\vartheta }_{1}dxd\rho dt=0,\forall {\vartheta }_1\in L^2\left(0,T;H_{{\Gamma }_0}^1\left(\Omega \times \left(0,1\right)\right)\right),$$

$${\int }_0^T{\int }_0^1{\int }_{{\Gamma }_1}(\tau z_{2_t}+z_{2\rho }){\vartheta }_{2}d\sigma d\rho dt=0,\forall {\vartheta }_2\in L^2\left(0,T;H^1\left({\Gamma }_1\times \left(0,1\right)\right)\right).$$

Uniqueness

Let $(u,v,z_1,z_2)$ and $(\tilde{u},\tilde{v},{\tilde{z}}_1,{\tilde{z}}_2)$ be two solutions to problem (6). Then $\left(U,V,Z_1,Z_2\right)=(u,v,z_1,z_2)-(\tilde{u},\tilde{v},{\tilde{z}}_1,{\tilde{z}}_2)$ verifies the following system of equations:

$$\left\{\begin{array}{ccc}\begin{array}{c}{U}_{tt}-\Delta U+{\mu }_1{g}_1\left(u_t\right)-{\mu }_1{g}_1\left({\tilde{u}}_t\right)\\ +{\mu }_2{g}_1\left(z_1(x,1,t)\right)-{\mu }_2{g}_1\left({\tilde{z}}_1(x,1,t)\right)=0,\end{array}& \begin{array}{c}\\ \mathrm{in}\end{array}& \begin{array}{c}\\ \Omega \times (0,\infty ),\end{array}\\ \begin{array}{c}{V}_{tt}+{\partial }_{\nu }U-{\Delta }_{T}V+{\mu }_1^{\prime }{g}_2\left(v_t\right)-{\mu }_1^{\prime }{g}_2\left({\tilde{v}}_t\right)\\ +{\mu }_2^{\prime }{g}_2\left(z_2(x,1,t)\right)-{\mu }_2^{\prime }{g}_2\left({\tilde{z}}_2(x,1,t)\right)=0,\end{array}& \begin{array}{c}\\ \mathrm{on}\end{array}& \begin{array}{c}\\ {\Gamma }_1\times (0,\infty ),\end{array}\\ \tau Z_{1_t}(x,\rho ,t)+Z_{1_{\rho }}(x,\rho ,t)=0,& \mathrm{in}& \Omega \times (0,1)\times (0,\infty ),\\ \tau Z_{2_t}(x,\rho ,t)+Z_{2_{\rho }}(x,\rho ,t)=0,& \mathrm{on}& {\Gamma }_1\times (0,1)\times (0,\infty ),\\ U=V,& \mathrm{on}& \Gamma \times (0,\infty ),\\ U=0,& \mathrm{on}& {\Gamma }_0\times (0,\infty ),\\ Z_1(x,0,t)=U_t(x,t),& \mathrm{in}& \Omega \times (0,\infty ),\\ Z_2(x,0,t)=V_t(x,t),& \mathrm{on}& {\Gamma }_1\times (0,\infty ),\\ \left(U\right(0),V(0\left)\right)=(0,0),& \mathrm{in}& \Omega \times \Gamma ,\\ (U_t\left(0\right),V_t\left(0\right))=(0,0),& \mathrm{in}& \Omega \times \Gamma ,\\ Z_1(x,\rho ,0)=0,& \mathrm{in}& \Omega \times (0,1),\\ Z_2(x,\rho ,0)=0,& \mathrm{on}& {\Gamma }_1\times (0,1).\end{array}\right.$$

(71)

Multiplying the first equation of (71) by $U_t,$ we have

$$\begin{array}{ccc}& & (U_{tt},U_t)-(\Delta U,U_t)+{\mu }_1(g_1\left(u_t\right)-g_1\left({\tilde{u}}_t\right),U_t)\hfill \\ & & +{\mu }_2(g_1\left(z_1(x,1,t)\right)-g_1\left({\tilde{z}}_1(x,1,t)\right),U_t)\hfill \\ & =& 0,\hfill \end{array}$$

(72)

Next, integrating over $\Omega ,$ we obtain

$$-(\Delta U,U_t)=(\nabla U,\nabla U_t)-{({\partial }_{\nu }U,V_t)}_{{\Gamma }_1},$$

then, (72) becomes

$$\begin{array}{ccc}& & (U_{tt},U_t)+(\nabla U,\nabla U_t)-{({\partial }_{\nu }U,V_t)}_{{\Gamma }_1}+{\mu }_1(g_1\left(u_t\right)-g_1\left({\tilde{u}}_t\right),U_t)\hfill \\ & & +{\mu }_2(g_1\left(z_1(x,1,t)\right)-g_1\left({\tilde{z}}_1(x,1,t)\right),U_t)\hfill \\ & =& 0,\hfill \end{array}$$

which is equivalent to

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}{∥U_t∥}^2+\frac{1}{2}\frac{d}{dt}{∥\nabla U∥}^2-{({\partial }_{\nu }U,V_t)}_{{\Gamma }_1}+{\mu }_1(g_1\left(u_t\right)-g_1\left({\tilde{u}}_t\right),U_t)\hfill \\ & & +{\mu }_2(g_1\left(z_1(x,1,t)\right)-g_1\left({\tilde{z}}_1(x,1,t)\right),U_t)\hfill \\ & =& 0.\hfill \end{array}$$

(73)

Multiplying the second equation of (71) by $V_t,$ we have

$$\begin{array}{ccc}& & {(V_{tt},V_t)}_{{\Gamma }_1}+{({\partial }_{\nu }U,V_t)}_{{\Gamma }_1}-{({\Delta }_{T}V,V_t)}_{{\Gamma }_1}+{\mu }_1^{\prime }{(g_2\left(v_t\right)-g_2\left({\tilde{v}}_t\right),V_t)}_{{\Gamma }_1}\hfill \\ & & +{\mu }_2^{\prime }{(g_2\left(z_2(x,1,t)\right)-g_2\left({\tilde{z}}_2(x,1,t)\right),V_t)}_{{\Gamma }_1}\hfill \\ & =& 0,\hfill \end{array}$$

(74)

Next, integrating over ${\Gamma }_1,$ we obtain

$$-{({\Delta }_{T}V,V_t)}_{{\Gamma }_1}={({\nabla }_{T}V,{\nabla }_T{V}_t)}_{{\Gamma }_1},$$

then (74) becomes

$$\begin{array}{ccc}& & {(V_{tt},V_t)}_{{\Gamma }_1}+{({\partial }_{\nu }U,V_t)}_{{\Gamma }_1}+{({\nabla }_{T}V,{\nabla }_T{V}_t)}_{{\Gamma }_1}+{\mu }_1^{\prime }{(g_2\left(v_t\right)-g_2\left({\tilde{v}}_t\right),V_t)}_{{\Gamma }_1}\hfill \\ & & +{\mu }_2^{\prime }{(g_2\left(z_2(x,1,t)\right)-g_2\left({\tilde{z}}_2(x,1,t)\right),V_t)}_{{\Gamma }_1}\hfill \\ & =& 0,\hfill \end{array}$$

which is equivalent to

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}{∥V_t∥}_{{\Gamma }_1}^2+\frac{1}{2}\frac{d}{dt}{∥{\nabla }_{T}V∥}_{{\Gamma }_1}^2+{({\partial }_{\nu }U,V_t)}_{{\Gamma }_1}+{\mu }_1^{\prime }{(g_2\left(v_t\right)-g_2\left({\tilde{v}}_t\right),V_t)}_{{\Gamma }_1}\hfill \\ & & +{\mu }_2^{\prime }{(g_2\left(z_2(x,1,t)\right)-g_2\left({\tilde{z}}_2(x,1,t)\right),V_t)}_{{\Gamma }_1}\hfill \\ & =& 0.\hfill \end{array}$$

(75)

Now, summing (73) and (75), we obtain

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}{∥U_t∥}^2+\frac{1}{2}\frac{d}{dt}{∥\nabla U∥}^2+\frac{1}{2}\frac{d}{dt}{∥V_t∥}_{{\Gamma }_1}^2+\frac{1}{2}\frac{d}{dt}{∥{\nabla }_{T}V∥}_{{\Gamma }_1}^2\hfill \\ & & +{\mu }_1(g_1\left(u_t\right)-g_1\left({\tilde{u}}_t\right),U_t)+{\mu }_1^{\prime }{\left(g_2\left(v_t\right)-g_2\left({\tilde{v}}_t\right),V_t\right)}_{{\Gamma }_1}\hfill \\ & & +{\mu }_2\left(g_1\left(z_1(x,1,t)\right)-g_1\left({\tilde{z}}_1(x,1,t)\right),U_t\right)\hfill \\ & & +{\mu }_2^{\prime }{\left(g_2\left(z_2(x,1,t)\right)-g_2\left({\tilde{z}}_2(x,1,t)\right),V_t\right)}_{{\Gamma }_1}\hfill \\ & =& 0.\hfill \end{array}$$

(76)

Similarly, multiplying the third and fourth equations of (71) by $Z_1$ and $Z_2$, respectively, and integrating over $\Omega \times \left(0,1\right)$ and ${\Gamma }_1\times \left(0,1\right),$ we obtain

$$\frac{\tau }{2}\frac{d}{dt}{\int }_0^1{∥Z_1\left(x,\rho ,t\right)∥}^{2}d\rho +\frac{1}{2}\left\{{∥Z_1\left(x,1,t\right)∥}^2-{∥U_t\left(x,t\right)∥}^2\right\}=0,$$

(77)

$$\frac{\tau }{2}\frac{d}{dt}{\int }_0^1{∥Z_2\left(x,\rho ,t\right)∥}_{{\Gamma }_1}^{2}d\rho +\frac{1}{2}\left\{{∥Z_2\left(x,1,t\right)∥}_{{\Gamma }_1}^2-{∥V_t\left(x,t\right)∥}_{{\Gamma }_1}^2\right\}=0.$$

(78)

From (76)–(78), and using the Cauchy–Schwarz inequality, we obtain

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}\left\{{∥U_t∥}^2+{∥\nabla U∥}^2+{∥V_t∥}_{{\Gamma }_1}^2+{∥{\nabla }_{T}V∥}_{{\Gamma }_1}^2\right.\hfill \\ & & \left.+\tau {\int }_0^1{∥Z_1\left(x,\rho ,t\right)∥}^{2}d\rho +\tau {\int }_0^1{∥Z_2\left(x,\rho ,t\right)∥}_{{\Gamma }_1}^{2}d\rho \right\}\hfill \\ & & +{\mu }_1(g_1\left(u_t\right)-g_1\left({\tilde{u}}_t\right),U_t)+{\mu }_1^{\prime }{\left(g_2\left(v_t\right)-g_2\left({\tilde{v}}_t\right),V_t\right)}_{{\Gamma }_1}\hfill \\ & =& -{\mu }_2\left(g_1\left(z_1(x,1,t)\right)-g_1\left({\tilde{z}}_1(x,1,t)\right),U_t\right)\hfill \\ & & -{\mu }_2^{\prime }{\left(g_2\left(z_2(x,1,t)\right)-g_2\left({\tilde{z}}_2(x,1,t)\right),V_t\right)}_{{\Gamma }_1}\hfill \\ & & +\frac{1}{2}{∥U_t\left(x,t\right)∥}^2+\frac{1}{2}{∥V_t\left(x,t\right)∥}_{{\Gamma }_1}^2\hfill \\ & \le & \frac{1}{2}{∥U_t\left(x,t\right)∥}^2+{∥g_1\left(z_1(x,1,t)\right)-g_1\left({\tilde{z}}_1(x,1,t)\right)∥}^2{∥U_t\left(x,t\right)∥}^2\hfill \\ & & +\frac{1}{2}{∥V_t\left(x,t\right)∥}_{{\Gamma }_1}^2+{∥g_2\left(z_2(x,1,t)\right)-g_2\left({\tilde{z}}_2(x,1,t)\right)∥}_{{\Gamma }_1}^2{∥V_t\left(x,t\right)∥}_{{\Gamma }_1}^2,\hfill \end{array}$$

Next, using condition (4) and Young’s inequality, we find

$$\begin{array}{ccc}& & \frac{1}{2}\frac{d}{dt}\left\{{∥U_t∥}^2+{∥\nabla U∥}^2+{∥V_t∥}_{{\Gamma }_1}^2+{∥{\nabla }_{T}V∥}_{{\Gamma }_1}^2\right.\hfill \\ & & \left.+\tau {\int }_0^1{∥Z_1\left(x,\rho ,t\right)∥}^{2}d\rho +\tau {\int }_0^1{∥Z_2\left(x,\rho ,t\right)∥}_{{\Gamma }_1}^{2}d\rho \right\}\hfill \\ & \le & c\left\{{∥U_t\left(x,t\right)∥}^2+{∥V_t\left(x,t\right)∥}_{{\Gamma }_1}^2+{∥Z_1\left(x,1,t\right)∥}^2+{∥Z_2\left(x,1,t\right)∥}_{{\Gamma }_1}^2\right\},\hfill \end{array}$$

where $c>0$. Then, integrating over $\left(0,t\right)$ and using Gronwall’s lemma, we conclude that

$$\begin{array}{ccc}& & {∥U_t∥}^2+{∥\nabla U∥}^2+{∥V_t∥}_{{\Gamma }_1}^2+{∥{\nabla }_{T}V∥}_{{\Gamma }_1}^2\hfill \\ & & +\tau {\int }_0^1{∥Z_1\left(x,\rho ,t\right)∥}^{2}d\rho +\tau {\int }_0^1{∥Z_2\left(x,\rho ,t\right)∥}_{{\Gamma }_1}^{2}d\rho \hfill \\ & =& 0,\hfill \end{array}$$

which implies $\left(U,V,Z_1,Z_2\right)=0.$

This finishes the proof of Theorem 1. □

3. Conclusions

In this article, we have proven the existence of a unique solution to a wave equation with dynamic Wentztell-type boundary conditions on a part of the boundary ${\Gamma }_1$ of the domain $\Omega$ with non-linear delays in non-linear dampings in $\Omega$ and on ${\Gamma }_1$, using the Faedo–Galerkin method.