Stability for coupled thermoelastic systems with nonlinear localized damping and Wentzell boundary conditions

Abstract

This paper is concerning with the study of stability involving a thermoelastic system with internal nonlinear localized damping. The main novelty of the paper is to introduce to the study of thermoelastic system the general Wentzell boundary conditions associated to the internal heat equation. This boundary condition takes into account that there is a boundary source of heat which depends on the heat flow along the boundary, the heat flux across the boundary, and the temperature at the boundary. The tools are the use of multipliers with the construction of appropriate perturbed energy functionals.

1 Introduction

Let $\Omega \subset \mathbb {R}^N$ be an open, bounded and connected set, $N\ge 2$, with smooth boundary $\Gamma =\Gamma _0\cup \Gamma _1$ such that ${\overline{\Gamma }}_0\cap {\overline{\Gamma }}_1=\emptyset$. In this paper we study the following problem

$$\begin{aligned}{} & {} \displaystyle {u_{tt}-c\Delta u+\text{ div }(\theta )+\rho (x) g(u_t)=0} \,\, \text{ in }\,\Omega \times (0,\infty ), \end{aligned}$$

(1)

$$\begin{aligned}{} & {} \theta _t-\Delta \theta +\text{ div }(u_t)=0\,\, \text{ in }\,\Omega \times (0,\infty ), \end{aligned}$$

(2)

$$\begin{aligned}{} & {} u=0\,\, \text{ on }\,\Gamma \times (0,\infty ), \end{aligned}$$

(3)

$$\begin{aligned}{} & {} \theta =0\,\, \text{ on }\,\Gamma _0\times (0,\infty ), \end{aligned}$$

(4)

$$\begin{aligned}{} & {} \theta _t-\alpha \theta -\beta \Delta _{\Gamma } \theta +\beta \frac{\partial \theta }{\partial \nu }=0\,\, \text{ on }\,\Gamma _1\times (0,\infty ), \end{aligned}$$

(5)

$$\begin{aligned}{} & {} u(x,0)=u_0(x), \, u_t(x,0)=u_1(x),\,\theta (x,0)=\theta _0(x)\,\, x\in \Omega , \end{aligned}$$

(6)

where $\Delta$ and $\Delta _{\Gamma }$ are, respectively, the Laplace and Laplace-Beltrami operators in the spatial variable; $\nu$ is the outward unit normal vector at $\Gamma$; c is a positive real number; $\alpha ,\beta :\Gamma _1\rightarrow \mathbb {R}$ are positive and continuous functions; $u_0,u_1$, and $\theta _0$ are the initial data; and $\rho$ is a nonnegative function responsible for the localized damping effect.

The problem (1, 2) is a n-dimensional version of the classical one-dimensional thermoelastic system

$$\begin{aligned}{} & {} \displaystyle {u_{tt}-u_{xx}+\gamma \theta _x=0} \,\, \text{ in }\,(0,L)\times (0,\infty ), \end{aligned}$$

(7)

$$\begin{aligned}{} & {} \theta _t-k\theta _{xx}+\gamma \theta _{xt} =0\,\, \text{ in }\,(0,L)\times (0,\infty ), \end{aligned}$$

(8)

where u is the displacement, $\theta$ is the temperature deviation from the reference temperature, and $\gamma$ and k are positive constants depending on the material. The system (7)–(8) was studied for instance by Dafermos [17], Liu and Zheng [33, 34] and Rivera [35]. In [33, 35] the authors proved that, even in the absence of damping term, the energy associated to the problem decays. The n-dimensional case was studied by Clark, San Gil Jutuca, and Milla Miranda [12] and Apolaya, Clark, and Feitosa [1]. In [12] the authors proved the exponential stability with the damping term acting on a boundary portion of the domain. In [1] they studied the system without damping, the authors also considered a time-dependent coefficient multiplying the Laplace operator.

More recently Braz e Silva, Clark, and Frota [6] proved the existence, uniqueness, and asymptotic behavior of global solutions for the following thermoelastic system with nonlocal nonlinearities under the acoustic boundary conditions

$$\begin{aligned}{} & {} \displaystyle {u_{tt}-c\Delta u+\lambda |u|^{\rho }u+(a\cdot \nabla )\theta =0} \,\, \text{ in }\,\Omega \times (0,\infty ), \end{aligned}$$

(9)

$$\begin{aligned}{} & {} \theta _t-\beta \left( \int _{\Omega }\theta \,dx\right) \Delta \theta +(a\cdot \nabla )u_t=0\,\, \text{ in }\,\Omega \times (0,\infty ), \end{aligned}$$

(10)

$$\begin{aligned}{} & {} u=0\,\, \text{ on }\,\Gamma _0\times (0,\infty ), \end{aligned}$$

(11)

$$\begin{aligned}{} & {} u_t+f_1\delta _{tt}+f_2\delta _t+f_3\delta =0\,\, \text{ on }\,\Gamma _0\times (0,\infty ), \end{aligned}$$

(12)

$$\begin{aligned}{} & {} \frac{\partial u}{\partial \nu }-\delta _t+\eta (\cdot ,u_t)=0\,\, \text{ on }\,\Gamma _1\times (0,\infty ), \end{aligned}$$

(13)

where $\beta ,f_1,f_2,f_3,$ and $\eta$ are known functions, $c,\rho ,$ and $\lambda$ are constants, and a is a known vector of $\mathbb {R}^N$. In [6], to prove the existence and uniqueness of solutions the authors employed the Faedo-Galerkin method and the energy method, respectively, with no restrictions on the geometry of the domain. To prove that the energy associated to the problem is uniformly stable, the authors used some usual restrictions on the geometry of the domain. Problems with acoustic boundary conditions can be found in [4, 5, 8, 20,21,22, 25, 32, 39, 43] and references therein. We highlight the work of Frota and Goldstein [22] which was the pioneer paper studying nonlinear problems.

On the other hand, the boundary condition (5) is associated to the following equation

$$\begin{aligned} Au+\beta \partial _n^a u+ \gamma u-q\beta \Delta _{\Gamma } u=0 \,\,\text{ on } \Gamma , \end{aligned}$$

(14)

where A is a second order uniformly elliptic operator defined by

$$\begin{aligned} Au =\sum _{i,j=1}^N\partial _i(a_{ij}(x)\partial _j u) = \nabla \cdot (a(x)\nabla )u, \end{aligned}$$

here $a=(a_{ij}(\cdot ))_{1\le i,j\le N}$ and $a_{ij}$ are real valued functions, $\partial _n^a u$ is the conormal derivative of u with respect to the matrix a, and $\gamma$ and $\beta$ are continuously differentiable functions. Problems with the boundary condition (14) has been studied by many authors. See [13,14,15,16, 18, 19] and references therein. They are called into the literature the general Wentzell boundary conditions (GWBC). Recently, Romanelli [36] called these the Goldstein–Wentzell boundary conditions (GWBC). See also the works of Cavalcanti, Lasiecka, and Toundykov [10, 11].

Concerning the GWBC we would like to cite the paper which is our main motivation to study the system (1–6). In [26], G. R. Goldstein gives new derivations of the heat and wave equations which incorporate the boundary conditions into the formulation of the problems. She makes several descriptions on classical boundary conditions as well as on the general Wentzell and dynamic boundary conditions. Our motivation is precisely Sect. 3 of the paper where she considered the heat equation and GWBC. For the reader’s convenience we rewrite the ideas of G. R. Goldstein here. It is well known that the linear heat equation on a domain $\Omega$ is given by

$$\begin{aligned} (\rho c \theta )_t= \kappa \Delta \theta +s, \end{aligned}$$

(15)

where $\theta (x,t)$ represents the temperature at position $x\in {\overline{\Omega }}$ at time $t\ge 0$; $\kappa$ is the thermal conductivity constant, $\rho$ is the density of the material, c is the heat capacity of the material, and s represents a heat source. We suppose that there exists a heat source acting on the boundary of the region $\Omega$. Moreover, we suppose that the source should depend on the heat flow along the boundary, the heat flux across the boundary and the temperature at the boundary. If we take it into account, then the boundary condition becomes

$$\begin{aligned} \theta _t=a(x)\Delta _{\Gamma }\theta -b(x)\frac{\partial \theta }{\partial \nu } +c(x)\theta \,\,\text{ on } \Gamma , \end{aligned}$$

(16)

where a, b,  and c are known functions. The Laplace-Beltrami operator describes the heat flux along the boundary and, since $c>0$, the term $c\theta$ represents a heat source on the boundary.

Therefore, observing (15, 16), the main goal of the present paper is to incorporate into the thermoelastic system the equation (5) which takes into account the heat flow along the boundary, the heat flux across the boundary, and the temperature at the boundary. The result extends the preview literature involving the thermoelastic system, because, to the best of our knowledge, it is the first concerning the GWBC associated to the heat equation. We would like to mention that the present paper extends the discussion started by Bras Silva, Clark, and Frota [6]. Indeed, [6] was the first paper concerning some dynamics on a boundary portion using the acoustic boundary conditions. But their boundary equation is associated to the internal wave equation (and $\delta$ models the boundary behaviour) while in our manuscript the boundary equation is associated to the internal heat equation (and it models the temperature at the boundary). Our work also extends in some direction the paper of Kasri [30] where a thermoelastic system with static Wentzell boundary conditions was studied. We highlight that in [30] the boundary equation also is associated to the internal wave equation.

The tools of our work are the use of multipliers with the construction of appropriate perturbed energy functionals. We consider that the function g satisfies the assumptions introduced by Lasiecka and Tataru [31]. Due to the localized damping effect and the presence of nonhomogeneos boundary conditions, there are some technical difficulties to overcome.

Finally, we cite the work of Frota, Medeiros, and Vicente [23] which studied problems with acoustic boundary conditions to non-locally reacting boundary. This boundary condition involves the Laplace–Beltrami operator and it is associated to the motion of the boundary. See also [2, 24, 27,28,29, 38, 40,41,42].

The paper is organized as follows. In Sect. 2 we present the notation and the existence theorem. In Sect. 3 we prove the stability result, the main theorem of the paper.

2 Notations and existence of solution

As described in the introduction, in this section we present the notations and an existence theorem. We suppose that the following assumptions hold.

Assumption 1. The function $\rho$ satisfies

$$\begin{aligned} \rho (x) \ge \rho _0 >0~\hbox { a.e. in } ~\omega , \end{aligned}$$

(17)

where $\omega$ is a neighborhood, in $\Omega$, of $\Gamma _1$.

Assumption 2. The function g is continuous and monotone increasing such that

$$\begin{aligned} \left\{ \begin{aligned}&g(s)s >0 \hbox { for all } s\ne 0,\\&c_1 |s| \le |g(s)| \le c_2 |s|\hbox { for all } |s| \ge 1, \end{aligned} \right. \end{aligned}$$

(18)

for some positive constants $c_1,c_2$.

We recall that Assumption 2 is the classical one introduced by Lasiecka and Tataru [31].

We denote by $\Vert \cdot \Vert _{L^2(\Omega )}$ the usual norm in the Hilbert space $L^2(\Omega )$ endowed with the inner product $(u,v)_{L^2(\Omega )}=\int _{\Omega }u(x)v(x)\,dx$. We consider $H_0^1(\Omega )$, which is a Hilbert space with the inner product

$$\begin{aligned} (u,v)_{H_0^1(\Omega )}=\int \limits _{\Omega }c \nabla u\cdot \nabla v\,dx. \end{aligned}$$

We also consider the subspace of $H^1(\Omega )$, denoted by V, as the closure of $C^1({\overline{\Omega }})$ such that $u_{|_{\Gamma _0}}=0$ in the strong topology of $H^1(\Omega )$, i.e.,

$$\begin{aligned} V=\overline{\{u\in C^1({\overline{\Omega }});\,u_{|_{\Gamma _0}}=0\}}^{H^1(\Omega )}. \end{aligned}$$

We know that the Poincaré inequality holds in V, thus there exists a positive constant $c_p$ such that

$$\begin{aligned} \int \limits _{\Omega } u^2\,dx\le c_p\int _{\Omega }|\nabla u|^2\,dx, \end{aligned}$$

for all $u\in V$. Therefore, the space V can be endowed with the norm, $\Vert \nabla \cdot \Vert _{L^2(\Omega )}$, induced by the inner product

$$\begin{aligned} (u,v)_V=\int \limits _{\Omega }\nabla u\cdot \nabla v\,dx, \end{aligned}$$

which is equivalent to usual norm of $H^1(\Omega )$. The dual space of V is denoted by $V'$.

Finally, we define

$$\begin{aligned} L^2_{\beta }(\Gamma _1) = \left\{ u:\Omega \rightarrow \mathbb {R};\,\ \displaystyle \int \limits _{\Gamma _1}\frac{1}{\beta }u^2 \,d\Gamma <\infty \right\} , \end{aligned}$$

which is endowed with the inner product

$$\begin{aligned} (u,v)_{L^2_{\beta }(\Gamma _1)}=\int \limits _{\Gamma _1}\frac{1}{\beta }uv\,d\Gamma , \end{aligned}$$

and norm

$$\begin{aligned} \Vert u\Vert _{L^2_{\beta }(\Gamma _1)} = \left( \displaystyle \int \limits _{\Gamma _1}\frac{1}{\beta }u^2\,d\Gamma \right) ^{1/2}. \end{aligned}$$

We denote by $\gamma _0: H^1(\Omega )\rightarrow H^{\frac{1}{2}}(\Gamma )$ the trace map of order zero and by $\gamma _1: H^1(\Omega )\rightarrow H^{-\frac{1}{2}}(\Gamma )$ the trace map of order one, i.e. $\gamma _1(\cdot )=\frac{\partial \cdot }{\partial \nu }$.

We define

$$\begin{aligned} \mathcal {H}= H_0^1(\Omega )\times L^2(\Omega ) \times L^2(\Omega )\times L^2_{\beta }(\Gamma _1) \end{aligned}$$

with the inner product and norm given by

$$\begin{aligned} ((u,v,\theta ,z),(r,s,\mu ,p))_{\mathcal {H}}=(u,r)_{H_0^1(\Omega )}+(v,s)_{L^2(\Omega )}+(\theta ,\mu )_{L^2(\Omega )}+(z,p)_{L_{\beta }^2(\Gamma _1)} \end{aligned}$$

and

$$\begin{aligned} \Vert (u,v,\theta ,z)\Vert _{\mathcal {H}}^2 = \Vert u\Vert _{H_0^1(\Omega )}^2 + \Vert v\Vert _{L^2(\Omega )}^2 + \Vert \theta \Vert ^2_{L^2(\Omega )} +\Vert z \Vert ^2_{L^2_{\beta }(\Gamma _1)}. \end{aligned}$$

Finally, we define the operator $\mathcal {A}:D(\mathcal {A})\subset \mathcal {H}\rightarrow \mathcal {H}$ by

$$\begin{aligned} \mathcal {A}\left( \begin{array}{c} u\\ v\\ \theta \\ z \end{array} \right) = \left( \begin{array}{c} v\\ c \Delta u -\text{ div }(\theta )-\rho (x)g(v)\\ \Delta \theta -\text{ div }(v)\\ \beta \Delta _{\Gamma }z+\alpha z -\beta \frac{\partial \theta }{\partial \nu } \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned}{} & {} D(\mathcal {A})= \Big \{ (u,v,\theta ,z)\in \mathcal {H}; \,v\in H_0^1(\Omega ), \,\theta \in V,\, c\Delta u -\text{ div }(\theta )-\rho (\cdot )g(v)\in L^2(\Omega ), \\{} & {} \Delta \theta -\text{ div }(v)\in L^2(\Omega ),\, \, \beta \Delta _{\Gamma }z+\alpha z -\beta \frac{\partial \theta }{\partial \nu }\in L^2_{\beta }(\Gamma _1),\, \gamma _0(\theta )=z \Big \}. \end{aligned}$$

Therefore, the problem (1–6) can be written as

$$\begin{aligned}{} & {} \frac{d}{dt} U (t) = \mathcal {A} U(t)\,\, \text{ in } (0,\infty ), \end{aligned}$$

(19)

$$\begin{aligned}{} & {} U(0)=U_0, \end{aligned}$$

(20)

where $U=(u,v,\theta ,\gamma _0(\theta ))^T$ and $U_0=(u_0,u_1,\theta _0,\gamma _0(\theta _0))^T\in D(\mathcal {A})$. To prove that (19–20) has solution, it suffices to show that the operator $\mathcal {A}-\delta I$ is maximal dissipative on $\mathcal {H}$ for some positive real number $\delta$. To prove that the operator is dissipative, we define $y_i=(u_i,v_i,\theta _i,z_i)^T\in D(\mathcal {A})$, $i=1,2$, $y=(u,v,\theta ,z)=y_1-y_2$. We observe that

$$\begin{aligned}{} & {} (\mathcal {A}y_1-\mathcal {A}y_2,y_1-y_2)_{\mathcal {H}}-\delta (y_1-y_2,y_1-y_2)_{\mathcal {H}} \\{} & {} \quad = \int \limits _{\Omega }c\nabla v\cdot \nabla v\,dx + \int \limits _{\Omega }[c \Delta u-\text{ div }(\theta )-\rho (x)(g(v_1)-g(v_2))] v\,dx \\{} & {} \quad + \int \limits _{\Omega }( \Delta \theta -\text{ div }(v))\theta \,dx + \int \limits _{\Gamma _1}\left( \beta \Delta _{\Gamma } \theta +\alpha \theta -\beta \frac{\partial \theta }{\partial \nu }\right) \theta \frac{1}{\beta }\,d\Gamma \\{} & {} \quad -\delta (y_1-y_2,y_1-y_2)_{\mathcal {H}} \le 0, \end{aligned}$$

for $\delta$ large enough. Thus, the operator $\mathcal {A}-\delta I$ is dissipative.

To show that $\mathcal {A}-\delta I$ is maximal dissipative it is sufficient to prove that the operator $\lambda I-\mathcal {A}$ is onto $\mathcal {H}$ for some $\lambda >\delta$. Thus, let $(x_1,x_2,x_3,x_4)$ be an arbitrary element of $\mathcal {H}$. We are going to prove that there exists $(u,v,\theta ,z)\in D(\mathcal {A})$ such that

$$\begin{aligned} (\lambda I-\mathcal {A}) \left( \begin{array}{c} u\\ v\\ \theta \\ z \end{array} \right) = \left( \begin{array}{c} x_1\\ x_2\\ x_3\\ x_4 \end{array} \right) , \end{aligned}$$

(21)

for some positive $\lambda$. We define $A:\mathcal {D}(A)\subset L^2(\Omega )\rightarrow L^2(\Omega )$ by

$$\begin{aligned} Au=-c \Delta u, \end{aligned}$$

where $\mathcal {D}(A)=H_0^1(\Omega )\cap H^2(\Omega )$. To deal with the heat equation, we define $B:\mathcal {D}(B)\subset L^2(\Omega )\rightarrow L^2(\Omega )$ by

$$\begin{aligned} B\theta =- \Delta \theta , \end{aligned}$$

where

$$\begin{aligned} D(B) = \left\{ \psi \in L^2(\Omega );\, \Delta \psi \in L^2(\Omega ),\, \psi _{|_{\Gamma _0}}=0,\, \frac{\partial \psi }{\partial \nu }_{|_{\Gamma _1}}=0 \right\} . \end{aligned}$$

Let $\mathcal {N}: L_{\beta }^2(\Gamma _1)\rightarrow H^1(\Omega )$ be the Neumann operator such that $\phi \mapsto \mathcal {N}\phi$, where $\mathcal {N}\phi$ is the solution of

$$\begin{aligned} \left\{ \begin{array}{lcl} \Delta \mathcal {N}\phi =0 &{} \text{ in } &{}\Omega ,\\ \displaystyle \frac{\partial \mathcal {N}\phi }{\partial \nu }=\phi &{} \text{ on } &{}\Gamma _1,\\ \mathcal {N}\phi =0 &{} \text{ on } &{}\Gamma _0. \end{array} \right. \end{aligned}$$

Therefore, (21) becomes

$$\begin{aligned} \begin{array}{l} \lambda v+\frac{1}{\lambda }Av+\text{ div }(\theta )+\rho (x)g(v)=x_2-\frac{1}{\lambda }Ax_1\\ \lambda \theta +B \left( \theta +\mathcal {N} \left( \frac{\beta \Delta _{\Gamma }\theta +\alpha \theta -\lambda \theta }{\beta } \right) \right) +\text{ div }(v) = x_3-B\mathcal {N}\left( \frac{x_4}{\beta }\right) . \end{array} \end{aligned}$$

(22)

Moreover, this problem can be written as

$$\begin{aligned} (F+C+M)\left( \begin{array}{c} v\\ \theta \end{array} \right) = \left( \begin{array}{c} x_2-\frac{1}{\lambda }Ax_1\\ x_3-B\mathcal {N}\left( \frac{x_4}{\beta }\right) \end{array} \right) , \end{aligned}$$

(23)

where $F:H_0^1(\Omega )\times V\rightarrow H^{-1}(\Omega )\times V'$ is the duality mapping of $L^2(\Omega )\times L^2(\Omega )$ given by

$$\begin{aligned} F \left( \begin{array}{c} v\\ \theta \end{array} \right) = \left( \begin{array}{cc} \frac{1}{\lambda }A &{} 0\\ 0 &{} B \end{array} \right) \left( \begin{array}{c} v\\ \theta \end{array} \right) , \end{aligned}$$

$C:H_0^1(\Omega )\times V\rightarrow H^{-1}(\Omega )\times V'$ is the bounded, hemicontinuous and monotone operator defined by

$$\begin{aligned} C \left( \begin{array}{c} v\\ \theta \end{array} \right) = \left( \begin{array}{ccc} \lambda &{} &{}\text{ div }\\ \text{ div } &{} &{}\lambda +\mathcal {N} \left( \frac{\beta \Delta _{\Gamma }\cdot +\alpha \cdot -\lambda \cdot }{\beta } \right) \end{array} \right) \left( \begin{array}{c} v\\ \theta \end{array} \right) , \end{aligned}$$

and $M:H_0^1(\Omega )\times V\rightarrow H^{-1}(\Omega )\times V'$ is the maximal monotone (see [7]) operator given by

$$\begin{aligned} M\left( \begin{array}{c} v\\ \theta \end{array} \right) = \left( \begin{array}{c} \rho (x)g(v)\\ 0 \end{array} \right) . \end{aligned}$$

(24)

Since M is maximal monotone and C is monotone and hemicontinuous, we can use Corollary 1.3 of Barbu [3, page 48] to conclude that $C+M$ is maximal monotone. From this and as F is a duality mapping, we can use Theorem 1.2 of Barbu [3, page 39] to infer that $R(F+C+M)$ is all of $H^{-1}(\Omega )\times V'$. Thus, there exists $(v,\theta )\in H_0^1(\Omega )\times V$ such that (23) holds. Consequently, (22) also holds. Defining $u=\frac{x_1+v}{\lambda }$ and $z=\gamma _0(\theta )$, we have that $(u,v,\theta ,z)\in D(\mathcal {A})$ satisfies (21). Therefore, $\mathcal {A}-\delta I$ is maximal dissipative. From nonlinear semigroup theory, there exists a unique solution $U\in C([0,T];D(\mathcal {A}))$ of (19)–(20) for any $T>0$ finite (see Showalter [37]). Summarizing, we have the following result.

Theorem 2.1

(Existence and uniqueness) Assume that Assumptions 1 and 2 hold. If $(u_0,u_1,\theta _0,\gamma _0(\theta _0))\in D(\mathcal {A})$, then (1–6) has a unique solution $(u,u_t,\theta ,\gamma _0(\theta )) \in C([0, T]; D(\mathcal {A}))$, for all $T>0$.

3 Stability

In this section, we prove the main result. We start by defining the energy associated to the problem (1–6) by

$$\begin{aligned} E(t) = \frac{1}{2} \left( \int _{\Omega }u_t^2\,dx + c\int \limits _{\Omega }|\nabla u|^2\,dx + \int \limits _{\Omega }\theta ^2\,dx + \int \limits _{\Gamma _1}\frac{1}{\beta }\theta ^2\,d\Gamma \right) . \end{aligned}$$

(25)

To prove the stability it is necessary to make more two assumptions.

Assumption 3. Let $x_0$ be a fixed point of $\mathbb {R}^N$. We define

$$\begin{aligned} m(x)=(x-x_0)\cdot \nu , \end{aligned}$$

for all $x\in \mathbb {R}^N$. We consider that the boundary $\Gamma$ of $\Omega$ is given by

$$\begin{aligned} \Gamma _0 = \{ x\in \Gamma ;\,m\cdot \nu < 0 \} \quad \text{ and } \quad \Gamma _1 = \{ x\in \Gamma ;\,m\cdot \nu \ge 0 \}. \end{aligned}$$

Since the trace map $\gamma _0$ is continuous, there exists a positive constant $c_{tr}$ such that

$$\begin{aligned} \int \limits _{\Gamma _1} \theta ^2\,d\Gamma \le c_{tr} \int \limits _{\Omega }|\nabla \theta |^2\,dx, \end{aligned}$$

(26)

for all $\theta \in V$.

Assumption 4. We suppose that $\alpha$ and $\beta$ satisfy

$$\begin{aligned} \max _{x\in {\overline{\Gamma }}_1}|\alpha (x)| \le \frac{\min \limits _{x\in {\overline{\Gamma }}_1}|\beta (x)|}{2c_{tr}}. \end{aligned}$$

(27)

Lemma 3.1

Suppose that Assumptions 1, 2,  and 3 hold. Suppose that $(u_0,u_1,\theta _0,\gamma _0(\theta _0))\in D(\mathcal {A})$ and let $(u,u_t,\theta ,\gamma _0(\theta ))$ be the solution of (1–6) given by Theorem 2.1 and E(t) the energy defined in (25). Then, we have

$$\begin{aligned} E'(t) +\int \limits _{\Omega }|\nabla \theta |^2\,dx +\int \limits _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma -\int \limits _{\Gamma _1}\frac{\alpha }{\beta }\theta ^2\,d\Gamma +\int \limits _{\Omega }\rho (x) g(u_t)u_t\,dx =0, \nonumber \\ \end{aligned}$$

(28)

for all $t\ge 0$. Moreover, if Assumption 4 holds, then

$$\begin{aligned} E'(t)\le 0, \end{aligned}$$

(29)

for all $t\ge 0$.

Proof

Multiplying (1) by $u_t$ and integrating over $\Omega$, we have

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \left( \int _{\Omega }u_t^2\,dx + c\int \limits _{\Omega }|\nabla u|^2\,dx \right) +\int \limits _{\Omega }u_t\text{ div }(\theta )\,dx +\int \limits _{\Omega }\rho (x) g(u_t)u_t\,dx =0. \nonumber \\ \end{aligned}$$

(30)

Multiplying (2) by $\theta$ and integrating over $\Omega$, we obtain

$$\begin{aligned} \frac{1}{2} \frac{d}{dt} \int \limits _{\Omega }\theta ^2\,dx + \int \limits _{\Omega }|\nabla \theta |^2\,dx -\int \limits _{\Gamma _1}\frac{\partial \theta }{\partial \nu }\theta \,d\Gamma +\int \limits _{\Omega }\theta \text{ div }(u_t)\,dx =0. \end{aligned}$$

(31)

From (5), we infer

$$\begin{aligned} -\int \limits _{\Gamma _1}\frac{\partial \theta }{\partial \nu }\theta \,d\Gamma&= \int \limits _{\Gamma _1}\frac{1}{\beta } \left( \theta _t -\beta \Delta _{\Gamma } \theta -\alpha \theta \right) \theta \,d\Gamma \nonumber \\&= \frac{1}{2} \frac{d}{dt} \int \limits _{\Gamma _1}\frac{1}{\beta }\theta ^2\,d\Gamma +\int \limits _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma -\int \limits _{\Gamma _1}\frac{\alpha }{\beta }\theta ^2\,d\Gamma . \end{aligned}$$

(32)

Moreover, we also have

$$\begin{aligned} \int \limits _{\Omega }u_t\text{ div }(\theta )\,dx = -\int \limits _{\Omega }\theta \text{ div }(u_t)\,dx. \end{aligned}$$

(33)

Combining (30–33), we infer

$$\begin{aligned} E'(t) +\int \limits _{\Omega }|\nabla \theta |^2\,dx +\int \limits _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma -\int \limits _{\Gamma _1}\frac{\alpha }{\beta }\theta ^2\,d\Gamma +\int \limits _{\Omega }\rho (x) g(u_t)u_t\,dx =0.\nonumber \\ \end{aligned}$$

(34)

Using the inequality (26), we have

$$\begin{aligned} \int \limits _{\Gamma _1}\frac{\alpha }{\beta }\theta ^2\,d\Gamma \le \frac{c_{tr}\max \limits _{x\in {\overline{\Gamma }}_1}|\alpha (x)|}{\min \limits _{x\in {\overline{\Gamma }}_1}|\beta (x)|}\int _{\Gamma _1}|\nabla \theta |^2\,dx. \end{aligned}$$

(35)

Thus

$$\begin{aligned}{} & {} E'(t) + \left( \frac{1}{2}-\frac{c_{tr}\max \limits _{x\in {\overline{\Gamma }}_1}|\alpha (x)|}{\min \limits _{x\in {\overline{\Gamma }}_1}|\beta (x)|}\right) \int _{\Omega }|\nabla \theta |^2\,dx \nonumber \\{} & {} \quad +\int \limits _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma +\int _{\Omega }\rho (x) g(u_t)u_t\,dx \le 0. \end{aligned}$$

(36)

From (34), (36), Assumptions 2 and 4, we conclude the proof. $\square$

Now, for each $\varepsilon >0$, we define the perturbed energy by

$$\begin{aligned} E_{\varepsilon }(t)=E(t)+\varepsilon \Psi (t), \end{aligned}$$

(37)

where

$$\begin{aligned} \Psi (t)=2\int _{\Omega }u_t m\cdot \nabla u\,dx +(N-1)\int _{\Omega }u_t u\,dx. \end{aligned}$$

(38)

Our decay result follows the ideas of Lasiecka and Tataru [31] which gives us general decay rates. This idea was used by many authors, see for instance Cavalcanti, Domingos Cavalcanti and Lasiecka [9], where one can find examples of explicit decay rates. It is well known that, thanks to Assumption 2 it is possible to build a concave, strictly increasing function $\varphi$ such that $\varphi \left( 0\right) =0$ and

$$\begin{aligned} \varphi \left( sg(s)\right) \ge s^{2}+ g^{2}(s),\text {\ for } |s|<1. \end{aligned}$$

(39)

We define

$$\begin{aligned} {\tilde{\varphi }}(\cdot )=\varphi \Big (\frac{\cdot }{meas\left( \omega \times (0,T)\right) }\Big ). \end{aligned}$$

(40)

Since ${\tilde{\varphi }}$ is monotone increasing, we have that $MI+{\tilde{\varphi }}$ is invertible for all $M\ge 0$. We define

$$\begin{aligned} p(x)=(MI+{\tilde{\varphi }})^{-1}\left( Lx\right) , \end{aligned}$$

(41)

where $L=({C\,meas(\omega \times (0,T))})^{-1}$ and C is a specific positive constant. The function p is positive, continuous and strictly increasing with $p(0)=0$. We also consider the function

$$\begin{aligned} q(x)=x-(I+p)^{-1}\left( x\right) . \end{aligned}$$

Finally, let S(t) be the solution of the following ordinary differential equation

$$\begin{aligned} \frac{d}{dt}S(t)+q(S(t))=0,\,S(0)=E(0). \end{aligned}$$

(42)

Theorem 3.1

(Stability) Assume that Assumptions 1, 2, 3,  and 4 hold. Suppose that $(u_0,u_1,\theta _0,\gamma _0(\theta _0))\in D(\mathcal {A})$ and let $(u,u_t,\theta ,\gamma _0(\theta ))$ be the solution of (1–6) given by Theorem 2.1, then there exists a $T_0>0$ such that for any $T>T_0$ the energy satisfies

$$\begin{aligned} E(t)\le S\Big (\frac{t}{T}-1 \Big ), \end{aligned}$$

for all $t>T$, with $\lim \limits _{t\rightarrow \infty }S(t)=0$, decreasing monotonically (S(t) is the solution of (42)).

Proof

Taking the derivative of $E_{\varepsilon }$, we have

$$\begin{aligned} E_{\varepsilon }'(t)&\le -\frac{1}{2}\int \limits _{\Omega }|\nabla \theta |^2\,dx -\int _{\Omega }\rho (x) g(u_t)u_t\,dx\nonumber \\&\quad +\varepsilon \int \limits _{\Omega }u_{tt} [2 m\cdot \nabla u+(N-1)u]\,dx\nonumber \\ {}&\quad +\varepsilon \int \limits _{\Omega }u_t [2 m\cdot \nabla u_t+(N-1)u_t]\,dx. \end{aligned}$$

(43)

Since u is a solution of (1–6), we have

$$\begin{aligned} \int \limits _{\Omega }u_{tt} u\,dx&= \int \limits _{\Omega }[c \Delta u-\text{ div }(\theta ) -\rho (x)g(u_t)]u\,dx\nonumber \\&= -c\int \limits _{\Omega }|\nabla u|^2\,dx -\int \limits _{\Omega } u\,\text{ div }(\theta )\,dx -\int \limits _{\Omega }\rho (x)g(u_t)u\,dx. \end{aligned}$$

(44)

Moreover

$$\begin{aligned} 2\int \limits _{\Omega }u_t m\cdot \nabla u_t\,dx = -N\int \limits _{\Omega }u_t^2\,dx +\int \limits _{\Gamma } m\cdot \nu u_t^2\,d\Gamma . \end{aligned}$$

(45)

Observing the identity

$$\begin{aligned} 2\int \limits _{\Omega }\Delta u \,m\cdot \nabla u\,dx =(N-2)\int \limits _{\Omega }|\nabla u|^2\,dx +2\int \limits _{\Gamma }\frac{\partial u}{\partial \nu } m\cdot \nabla u\,d\Gamma -\int \limits _{\Gamma }m\cdot \nu |\nabla u|^2\,d\Gamma \end{aligned}$$

and that u is a solution of (1–6), we infer

$$\begin{aligned}&2\int \limits _{\Omega }u_{tt}\, m\cdot \nabla u\,dx \\&\quad =c(N-2)\int _{\Omega }|\nabla u|^2\,dx +2c\int \limits _{\Gamma }\frac{\partial u}{\partial \nu } m\cdot \nabla u\,d\Gamma \\&\qquad -c\int \limits _{\partial \Omega }m\cdot \nu |\nabla u|^2\,d\Gamma -2\int _{\Omega } \text{ div }(\theta )\, m\cdot \nabla u\,dx -2\int _{\Omega }\rho (x)g(u_t)\,m\cdot \nabla u\,dx. \end{aligned}$$

As $u=0$ on $\Gamma$, it holds

$$\begin{aligned} |\nabla u|^2=\left| \frac{\partial u}{\partial \nu }\right| ^2 \,\,\,\text{ and }\,\,\, \frac{\partial u}{\partial \nu }m\cdot \nabla u =m\cdot \nu \left( \frac{\partial u}{\partial \nu }\right) ^2 \text{ on } \Gamma . \end{aligned}$$

Therefore

$$\begin{aligned} 2\int \limits _{\Omega }u_{tt}\, m\cdot \nabla u\,dx&=c(N-2)\int \limits _{\Omega }|\nabla u|^2\,dx +c\int _{\Gamma }m\cdot \nu \left( \frac{\partial u}{\partial \nu }\right) ^2\,d\Gamma \nonumber \\&\quad -2\int \limits _{\Omega } \text{ div }(\theta )\, m\cdot \nabla u\,dx -2\int \limits _{\Omega }\rho (x)g(u_t)\,m\cdot \nabla u\,dx. \end{aligned}$$

(46)

Using (44–46) into (43), we obtain

$$\begin{aligned} E_{\varepsilon }'(t)&\le -\frac{1}{2}\int _{\Omega }|\nabla \theta |^2\,dx -\int _{\Omega }\rho (x) g(u_t)u_t\,dx\nonumber \\&\quad +\varepsilon \left[ c(N-2)\int _{\Omega }|\nabla u|^2\,dx +c\int _{\Gamma }m\cdot \nu \left( \frac{\partial u}{\partial \nu }\right) ^2\,d\Gamma \right. \nonumber \\&\quad -2\int _{\Omega } \text{ div }(\theta )\, m\cdot \nabla u\,dx -2\int _{\Omega }\rho (x)g(u_t)\,m\cdot \nabla u\,dx -c(N-1)\int _{\Omega }|\nabla u|^2\,dx\nonumber \\&\quad \left. -(N-1)\int _{\Omega } u\text{ div }(\theta )\,dx -(N-1)\int _{\Omega }\rho (x)g(u_t)u\,dx -\int _{\Omega }u_t^2\,dx \right] . \end{aligned}$$

(47)

Now, we are going to estimate the term $c\int _{\Gamma }m\cdot \nu \left( \frac{\partial u}{\partial \nu }\right) ^2\,d\Gamma$. We consider ${\hat{\omega }}$ a neighborhood of $\Gamma _1$ in $\Omega$ such that

$$\begin{aligned} \overline{{\hat{\omega }}}\cap \Omega \subset \omega . \end{aligned}$$

Let $h\in (W^{1,\infty }(\Omega ))^N$ be a vector field such that

$$\begin{aligned} \left\{ \begin{array}{l} h=\nu \quad \text{ on } \Gamma _1\\ h\cdot \nu \ge 0,\quad \text{ a. } \text{ e. } \text{ on } \Gamma \\ h=0\quad \text{ in } \Omega \setminus {\hat{\omega }}. \end{array} \right. \end{aligned}$$

We define

$$\begin{aligned} E_1(t) = 2\int \limits _{\Omega }u_th\cdot \nabla u\,dx. \end{aligned}$$

(48)

Thus

$$\begin{aligned}&E_1'(t) = 2\int \limits _{\Omega }u_th\cdot \nabla u_t\,dx +2\int _{\Omega }u_{tt}h\cdot \nabla u\,dx\nonumber \\&\quad = -\int \limits _{\Omega }\text{ div }(h) \,u_t^2\,dx +\int \limits _{\Gamma }h\cdot \nu \,u_t^2\,d\Gamma +2\int \limits _{\Omega }[c\Delta u-\text{ div }(\theta )\nonumber \\&\quad -\rho (x)g(u_t)]h\cdot \nabla u\,dx. \end{aligned}$$

(49)

Using Gauss’ theorem and observing the definition of the vector field h, we have

$$\begin{aligned}{} & {} 2c\int \limits _{\Omega }\Delta u\,h\cdot \nabla u\,dx =c\int \limits _{\Omega }\text{ div }(h)|\nabla u|^2\,dx +c\int \limits _{\Gamma _1}\left( \frac{\partial u}{\partial \nu }\right) ^2\,d\Gamma \nonumber \\{} & {} \quad -2c\int \limits _{\Omega }\frac{\partial u}{\partial x_i}\frac{\partial h_j}{\partial x_i}\frac{\partial u}{\partial x_j}\,dx, \end{aligned}$$

(50)

where $h=(h_1,h_2,\ldots ,h_N)$. Combining (49) with (50), we infer

$$\begin{aligned}&c\int \limits _{\Gamma }m\cdot \nu \left( \frac{\partial u}{\partial \nu }\right) ^2\,d\Gamma \le c\int _{\Gamma _1}m\cdot \nu \left( \frac{\partial u}{\partial \nu }\right) ^2\,d\Gamma \le c M_1\int \limits _{\Gamma _1}\left( \frac{\partial u}{\partial \nu }\right) ^2\,d\Gamma \nonumber \\&\quad =M_1\left[ E_1'(t) +\int _{\Omega }\text{ div }(h)[u_t^2-c|\nabla u|^2]\,dx +2c\int \limits _{\Omega }\frac{\partial u}{\partial x_i}\frac{\partial h_j}{\partial x_i}\frac{\partial u}{\partial x_j}\,dx \right. \nonumber \\&\quad \left. +2\int \limits _{\Omega }\text{ div }(\theta )\,h\cdot \nabla u\,dx +2\int \limits _{\Omega }\rho (x)g(u_t)\,h\cdot \nabla u\,dx \right] , \end{aligned}$$

(51)

where $M_1=\max \limits _{x\in {\overline{\Omega }}}|m(x)|$. From (47) and (51), we obtain

$$\begin{aligned}&E_{\varepsilon }'(t) \le -\frac{1}{2}\int _{\Omega }|\nabla \theta |^2\,dx -\int _{\Omega }\rho (x) g(u_t)u_t\,dx +\varepsilon \left\{ c(N-2)\int \limits _{\Omega }|\nabla u|^2\,dx \right. \nonumber \\&\quad +M_1\left[ E_1'(t) +\int _{\Omega }\text{ div }(h)[u_t^2-c|\nabla u|^2]\,dx +2c\int \limits _{\Omega }\frac{\partial u}{\partial x_i}\frac{\partial h_j}{\partial x_i}\frac{\partial u}{\partial x_j}\,dx \right. \nonumber \\&\quad \left. +2\int \limits _{\Omega }\text{ div }(\theta )\,h\cdot \nabla u\,dx +2\int _{\Omega }\rho (x)g(u_t)\,h\cdot \nabla u\,dx \right] \nonumber \\&\quad -2\int \limits _{\Omega } \text{ div }(\theta )\, m\cdot \nabla u\,dx -2\int _{\Omega }\rho (x)g(u_t)\,m\cdot \nabla u\,dx -c(N-1)\int \limits _{\Omega }|\nabla u|^2\,dx\nonumber \\&\quad \left. -(N-1)\int \limits _{\Omega } u\,\text{ div }(\theta )\,dx -(N-1)\int \limits _{\Omega }\rho (x)g(u_t)u\,dx -\int \limits _{\Omega }u_t^2\,dx \right\} . \end{aligned}$$

(52)

The next step is to estimate the term $\int _{{\hat{\omega }}}|\nabla u|^2\,dx$. Thus, we define a function $\eta :{\overline{\Omega }}\rightarrow \mathbb {R}$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} \eta =0 &{} \text{ a. } \text{ e. } \text{ in } \Omega \setminus \omega \\ \eta =1 &{} \text{ a. } \text{ e. } \text{ in } {\hat{\omega }}\\ 0\le \eta \le 1\\ \frac{|\nabla \eta |^2}{\eta }\in L^{\infty }(\omega ). \end{array} \right. \end{aligned}$$

We also define

$$\begin{aligned} E_2(t) = \int \limits _{\Omega }\eta \,u_tu\,dx. \end{aligned}$$

Taking the derivative of $E_2$, we have

$$\begin{aligned} c\int \limits _{{\hat{\omega }}}|\nabla u|^2\,dx&\le -2E_2'(t) +2\int \limits _{\Omega }\eta \,u_t^2\,dx +c\int \limits _{\Omega }\frac{|\nabla \eta |^2}{\eta } u^2\,dx\\&\quad -2\int _{\Omega }\eta \,u \,\text{ div }(\theta )\,dx +2\int \limits _{\Omega }\eta \,u \rho (x)g(u_t)\,dx. \end{aligned}$$

For each $\lambda >0$, we have

$$\begin{aligned} \int \limits _{\Omega }\eta \,u \,\text{ div }(\theta )\,dx \le C(\lambda )\int _{\Omega }|\nabla \theta |^2\,dx +\lambda E(t). \end{aligned}$$

Thus

$$\begin{aligned}&c\int \limits _{{\hat{\omega }}}|\nabla u|^2\,dx \le -2E_2'(t) +2\int _{\Omega }\eta \,u_t^2\,dx +c\int _{\Omega }\frac{|\nabla \eta |^2}{\eta } u^2\,dx +2\int \limits _{\Omega }\eta \,u \rho (x)g(u_t)\,dx\nonumber \\&\quad +C(\lambda )\int \limits _{\Omega }|\nabla \theta |^2\,dx +\lambda E(t). \end{aligned}$$

(53)

On the other hand, we have

$$\begin{aligned}&2\int \limits _{\Omega }\text{ div }(\theta )\,h\cdot \nabla u\,dx +2\int \limits _{\Omega }\text{ div }(\theta )\,h\cdot \nabla u\,dx -(N-1)\int \limits _{\Omega } u\,\text{ div }(\theta )\,dx\nonumber \\&\quad \le C(\lambda )\int \limits _{\Omega }|\nabla \theta |^2\,dx +\lambda E(t) \end{aligned}$$

(54)

and

$$\begin{aligned}{} & {} 2M_1\int \limits _{\Omega }\rho (x)g(u_t)\,h\cdot \nabla u\,dx +2\int \limits _{\Omega }\rho (x)g(u_t)\,m\cdot \nabla u\,dx\nonumber \\{} & {} \quad +(N-1)\int \limits _{\Omega }\rho (x)g(u_t)u\,dx \nonumber \\{} & {} \quad \le C(\lambda )\int \limits _{\Omega }\rho (x)g^2(u_t)\,dx +\lambda E(t). \end{aligned}$$

(55)

Therefore, (52–55) give

$$\begin{aligned} E_{\varepsilon }'(t)\le & {} -\left( \frac{1}{2}-C(\lambda )\varepsilon \right) \int \limits _{\Omega }|\nabla \theta |^2\,dx -\int \limits _{\Omega }\rho (x) g(u_t)u_t\,dx -\varepsilon \left[ 2-C\lambda \right] E(t) \nonumber \\{} & {} +M_1\varepsilon \left[ E_1'(t)-2E_2'(t)\right] +2M_0\varepsilon \int \limits _{\Omega }\eta u_t^2\,dx\nonumber \\{} & {} +C\varepsilon \int \limits _{\Omega }u^2\,dx +\varepsilon C(\lambda )\int _{\Omega }\rho (x)g^2(u_t)\,dx. \end{aligned}$$

(56)

Defining

$$\begin{aligned}{} & {} {\tilde{\Psi }}(t)=\Psi (t)+M_1E_1(t)-2M_1E_2(t), \end{aligned}$$

(57)

$$\begin{aligned}{} & {} {\tilde{E}}_{\varepsilon }(t)=E(t)+\varepsilon {\tilde{\Psi }}(t), \end{aligned}$$

(58)

and choosing $\lambda$ small enough, we infer

$$\begin{aligned}{} & {} {\tilde{E}}_{\varepsilon }'(t) +\varepsilon C_2 E(t) \nonumber \\{} & {} \quad \le -\left( \frac{1}{2}-C_1\varepsilon \right) \int \limits _{\Omega }|\nabla \theta |^2\,dx +C\varepsilon \int \limits _{\Omega }u^2\,dx +\varepsilon C\int _{\Omega }\rho (x)[u_t^2+g^2(u_t)]\,dx. \nonumber \\ \end{aligned}$$

(59)

It is not difficult to prove that there exists a positive constant ${\tilde{C}}$ such that

$$\begin{aligned} |{\tilde{E}}_{\varepsilon }(t)-E(t)|\le {\tilde{C}}\varepsilon E(t), \end{aligned}$$

(60)

for all $t\ge 0$ and for $\varepsilon >0$ small enough.

Integrating (59) from 0 to T and observing (60), we have

$$\begin{aligned}{} & {} (1-{\tilde{C}}\varepsilon )E(T)+\varepsilon C_2 \int _0^T E(t)\,dt\nonumber \\{} & {} \quad \le (1+{\tilde{C}}\varepsilon )E(0) -\left( \frac{1}{2}-C_1\varepsilon \right) \int \limits _0^T\int _{\Omega }|\nabla \theta |^2\,dx\,dt \nonumber \\{} & {} \qquad +C\varepsilon \int _0^T\int _{\Omega }u^2\,dx\,dt +\varepsilon C\int _0^T\int _{\Omega }\rho (x)[u_t^2+g^2(u_t)]\,dx\,dt. \end{aligned}$$

(61)

Since E(t) is decreasing, we have that

$$\begin{aligned} TE(T)\le \int _0^TE(t)\,dt. \end{aligned}$$

Thus, we infer

$$\begin{aligned}{} & {} (1+\varepsilon (C_2T-{\tilde{C}}))E(T)\nonumber \\ {}{} & {} \quad \le (1+{\tilde{C}}\varepsilon )E(0) -\left( \frac{1}{2}-C_1\varepsilon \right) \int \limits _0^T\int \limits _{\Omega }|\nabla \theta |^2\,dx\,dt \nonumber \\{} & {} \qquad +C\varepsilon \int _0^T\int \limits _{\Omega }u^2\,dx\,dt +\varepsilon C\int _0^T\int \limits _{\Omega }\rho (x)[u_t^2+g^2(u_t)]\,dx\,dt. \end{aligned}$$

(62)

On the other hand, Lemma 3.1 gives us

$$\begin{aligned}{} & {} E(0) = E(T) +\int \limits _0^T\int _{\Omega }|\nabla \theta |^2\,dx\,dt +\int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma \,dt \nonumber \\{} & {} \quad -\int \limits _0^T\int \limits _{\Gamma _1}\frac{\alpha }{\beta }\theta ^2\,d\Gamma \,dt +\int \limits _0^T\int _{\Omega }\rho (x) g(u_t)u_t\,dx\,dt. \end{aligned}$$

(63)

Substituting (63) into (62) and choosing $\varepsilon$ small enough, we obtain

$$\begin{aligned}&\varepsilon (C_2T-2{\tilde{C}})E(T) \le (1+{\tilde{C}}\varepsilon )\Big [\int _0^T\int _{\Omega }|\nabla \theta |^2\,dx\,dt +\int \limits _0^T\int \limits _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma \,dt\nonumber \\&\quad +\int \limits _0^T\int _{\Omega }\rho (x) g(u_t)u_t\,dx \,dt -\int _0^T\int \limits _{\Gamma _1}\frac{\alpha }{\beta }\theta ^2\,d\Gamma \,dt \Big ] \nonumber \\&\quad +C\varepsilon \int \limits _0^T\int _{\Omega }u^2\,dx\,dt +\varepsilon C\int _0^T\int \limits _{\Omega }\rho (x)[u_t^2+g^2(u_t)]\,dx\,dt. \end{aligned}$$

(64)

Choosing $T>0$ such that $C_2T-2{\tilde{C}}>0$ and using the continuity of the trace map, we have

$$\begin{aligned} E(T)\le & {} C \left[ \int \limits _0^T\int _{\Omega }|\nabla \theta |^2\,dx\,dt +\int _0^T\int \limits _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma \,dt \right. \nonumber \\{} & {} \left. +\varepsilon \int _0^T\int \limits _{\Omega }u^2\,dx\,dt +\varepsilon \int \limits _0^T\int \limits _{\Omega }\rho (x)[u_t^2+g^2(u_t)]\,dx\,dt \right] , \end{aligned}$$

(65)

for $\varepsilon >0$ small enough.

Now, we are going to estimate the low order term $\int _0^T\int \limits _{\Omega }u^2\,dx\,dt$. We claim that there exists a positive constant C such that

$$\begin{aligned}{} & {} \int _0^T\int \limits _{\Omega }u^2\,dx\,dt \nonumber \\{} & {} \quad \le C \left[ \int \limits _0^T\int _{\Omega }\{\rho (x)[u_t^2+g^2(u_t)]+|\nabla \theta |^2\}\,dx\,dt + \int _0^T\int \limits _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma \,dt \right] .\qquad \quad \end{aligned}$$

(66)

Indeed, suppose that (66) does not hold. Let $(u_{0k},u_{1k}, \theta _{0k},\gamma _0(\theta _{0k}))_{k\in \mathbb {N}}$ be a sequence of initial data and $(u_k,u_k',\theta _k,\gamma _0(\theta _k))_{k\in {\mathbb N}}$ the corresponding solutions of (1–6) such that

$$\begin{aligned} E_k(0)\le C, \end{aligned}$$

(67)

for all $k\in \mathbb {N}$, and one has

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{\int _0^T\int _{\Omega }u_k^2\,dx\,dt}{\int _0^T\int _{\Omega }\{\rho (x)[(u_k')^2+g^2(u_k')]+|\nabla \theta _k |^2\}\,dx\,dt + \int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }\theta _k|^2\,d\Gamma \,dt}=\infty ,\nonumber \\ \end{aligned}$$

(68)

where $'$ denotes the derivative with respect to the variable t and

$$\begin{aligned} E_k(t) = \frac{1}{2} \left( \int \limits _{\Omega }(u_k')^2\,dx + c\int \limits _{\Omega }|\nabla u_k|^2\,dx + \int \limits _{\Omega }\theta _k^2\,dx + \int \limits _{\Gamma _1}\frac{1}{\beta }\theta _k^2\,d\Gamma \right) . \end{aligned}$$

(69)

From (67) and (68), we have

$$\begin{aligned} \int _0^T\int \limits _{\Omega }\{\rho (x)[(u_k')^2+g^2(u_k')]+|\nabla \theta _k |^2\}\,dx\,dt + \int _0^T\int \limits _{\Gamma _1}|\nabla _{\Gamma }\theta _k|^2\,d\Gamma \,dt \rightarrow 0, \qquad \quad \end{aligned}$$

(70)

as $k\rightarrow \infty$. Observing (65), (67), and (70), we infer

$$\begin{aligned} E_k(t)\le C, \end{aligned}$$

(71)

for all $k\in \mathbb {N}$ and for all $t\ge 0$. Estimating (71) yields subsequences of $(u_k)_{k\in \mathbb {N}}$ and $(\theta _k)_{k\in \mathbb {N}}$, that we still denote in the same way, and functions $(u,\theta )$, such that

$$\begin{aligned}{} & {} u_k{\mathop {\rightharpoonup }\limits ^{*}} u \text{ in } L^{\infty }(0,T;H_0^1(\Omega )), \end{aligned}$$

(72)

$$\begin{aligned}{} & {} u_k'{\mathop {\rightharpoonup }\limits ^{*}} u' \text{ in } L^{\infty }(0,T;L^2(\Omega )), \end{aligned}$$

(73)

$$\begin{aligned}{} & {} \theta _k{\mathop {\rightharpoonup }\limits ^{*}} \theta \text{ in } L^{\infty }(0,T;L^2(\Omega )), \end{aligned}$$

(74)

as $k\rightarrow \infty$. Since $H_0^1(\Omega )$ is compactly embedded in $L^2(\Omega )$, from the Aubin–Lions Theorem, we have

$$\begin{aligned} u_k\rightarrow u \text{ in } L^{2}(0,T;L^2(\Omega )), \end{aligned}$$

(75)

as $k\rightarrow \infty$. On the other hand, from (70) we have

$$\begin{aligned} \theta _k\rightarrow 0 \text{ in } L^{2}(0,T;V), \end{aligned}$$

(76)

as $k\rightarrow \infty$. Thus (74) and (76) allow us to conclude that $\theta =0$.

At this point we are going to separate the proof into two cases.

Case $u\ne 0$.

For each $k\in \mathbb {N}$, $(u_k,\theta _k)$ is a solution of

$$\begin{aligned}{} & {} \displaystyle {u_k''-c\Delta u_k+\text{ div }(\theta _k)+\rho (x) g(u_k')=0} \,\, \text{ in }\,\Omega \times (0,T), \end{aligned}$$

(77)

$$\begin{aligned}{} & {} \theta _k'-\Delta \theta _k +\text{ div }(u_k')=0\,\, \text{ in }\,\Omega \times (0,T), \end{aligned}$$

(78)

$$\begin{aligned}{} & {} u_k=0\,\, \text{ on }\,\Gamma \times (0,T), \end{aligned}$$

(79)

$$\begin{aligned}{} & {} \theta _k=0\,\, \text{ on }\,\Gamma _0\times (0,T), \end{aligned}$$

(80)

$$\begin{aligned}{} & {} \theta _k'-\beta \Delta _{\Gamma } \theta _k +\beta \frac{\partial \theta _k}{\partial \nu }-\alpha \theta _k=0\,\, \text{ on }\,\Gamma _1\times (0,T). \end{aligned}$$

(81)

Taking to the limit, as $k\rightarrow \infty$, and observing (70) and (76), we obtain

$$\begin{aligned}{} & {} \displaystyle {u''-c\Delta u=0} \,\, \text{ in }\,\Omega \times (0,T), \end{aligned}$$

(82)

$$\begin{aligned}{} & {} u=0\,\, \text{ on }\,\Gamma \times (0,T), \end{aligned}$$

(83)

$$\begin{aligned}{} & {} u'=0\,\, \text{ on }\,\omega \times (0,T). \end{aligned}$$

(84)

Taking the derivative, with respect to t, and denoting by $v=u'$, we have

$$\begin{aligned}{} & {} \displaystyle {v''-c\Delta v=0} \,\, \text{ in }\,\Omega \times (0,T), \end{aligned}$$

(85)

$$\begin{aligned}{} & {} v=0\,\, \text{ on }\,\Gamma \times (0,T), \end{aligned}$$

(86)

$$\begin{aligned}{} & {} v=0\,\, \text{ on }\,\omega \times (0,T). \end{aligned}$$

(87)

Therefore uniqueness arguments give us that $v=u'=0$ in $\Omega \times (0,T)$. Thus $u''=0$ in $\Omega \times (0,T)$. Consequently (82)–(83) becomes

$$\begin{aligned}{} & {} \displaystyle {-\Delta u=0} \,\, \text{ in }\,\Omega \times (0,T), \end{aligned}$$

(88)

$$\begin{aligned}{} & {} u=0\,\, \text{ on }\,\Gamma \times (0,T). \end{aligned}$$

(89)

This allows us to conclude that $u=0$, which is a contradiction.

Case $u= 0$. For each $k\in \mathbb {N}$, we define

$$\begin{aligned}{} & {} c_k= \left( \int _0^T\int _{\Omega }u_k^2\,dx\,dt +\int _0^T\int _{\Omega }|\nabla \theta _k|^2\,dx\,dt +\int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }\theta _k|^2\,d\Gamma \,dt \right) ^{\frac{1}{2}}, \qquad \end{aligned}$$

(90)

$$\begin{aligned}{} & {} {\tilde{u}}_k=\frac{u_k}{c_k}, \quad \text{ and }\quad {\tilde{\theta }}_k=\frac{\theta _k}{c_k}. \end{aligned}$$

(91)

From (75) and (76), we infer

$$\begin{aligned} c_k\rightarrow 0, \end{aligned}$$

(92)

as $k\rightarrow \infty$. Moreover, we also have

$$\begin{aligned} \int _0^T\int _{\Omega }{\tilde{u}}_k^2\,dx\,dt +\int _0^T\int _{\Omega }|\nabla {\tilde{\theta }}_k|^2\,dx\,dt +\int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }{\tilde{\theta }}_k|^2\,d\Gamma \,dt =1, \end{aligned}$$

(93)

for all $k\in \mathbb {N}$.

The convergence (70) gives us

$$\begin{aligned} \int _0^T\int _{\Omega }\{\rho (x)[({\tilde{u}}_k')^2+\frac{g^2(u_k')}{c_k}]+|\nabla {\tilde{\theta }}_k |^2\}\,dx\,dt + \int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }{\tilde{\theta }}_k|^2\,d\Gamma \,dt \rightarrow 0, \nonumber \\ \end{aligned}$$

(94)

as $k\rightarrow \infty$. Therefore

$$\begin{aligned}{} & {} \sqrt{\rho }{\tilde{u}}_k'\rightarrow 0 \text{ in } L^2(0,T;L^2(\Omega )), \end{aligned}$$

(95)

$$\begin{aligned}{} & {} \sqrt{\rho }\frac{g(u_k')}{\sqrt{c_k}}\rightarrow 0 \text{ in } L^2(0,T;L^2(\Omega )), \end{aligned}$$

(96)

$$\begin{aligned}{} & {} {\tilde{\theta }}_k\rightarrow 0 \text{ in } L^2(0,T;V), \end{aligned}$$

(97)

as $k\rightarrow \infty$.

Adapting the proof of Lemma 3.1, it is possible to verify that

$$\begin{aligned} E_k(0)= & {} E_k(T) -\int _0^T\int _{\Omega }|\nabla \theta _k|^2\,dx\,dt -\int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }\theta _k|^2\,d\Gamma \,dt \nonumber \\{} & {} +\int _0^T\int _{\Gamma _1}\frac{\alpha }{\beta }\theta _k^2\,d\Gamma \,dt -\int _0^T\int _{\Omega }\rho (x) g(u_k')u_k'\,dx\,dt. \end{aligned}$$

(98)

On the other hand, analogously to (65), we infer

$$\begin{aligned} E_k(T)\le & {} C\left[ \int _0^T\int _{\Omega }u_k^2\,dx\,dt +\int _0^T\int _{\Omega }\rho (x)[(u_k')^2+g^2(u_k')]\,dx\,dt \right. \nonumber \\{} & {} \left. +\int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }\theta _k|^2\,d\Gamma \,dt +\int _0^T\int _{\Omega }|\nabla \theta _k|^2\,dx\,dt \right] . \end{aligned}$$

(99)

Now for each $k\in \mathbb {N}$, we define

$$\begin{aligned} {\tilde{E}}_k(t)=\frac{E_k(t)}{c_k}. \end{aligned}$$

Thus since $E_k(t)$ is decreasing and observing (98) and (99), we have

$$\begin{aligned} {\tilde{E}}_k(t) \le {\tilde{E}}_k(0) \le C +C\int _0^T\int _{\Omega }\rho (x)\left( ({\tilde{u}}_k')^2+\frac{g^2(u_k')}{c_k}\right) \,dx\,dt, \end{aligned}$$

(100)

for all $t\in [0,T]$. From (94) and (100), we conclude that

$$\begin{aligned} {\tilde{E}}_k(t) \le C, \end{aligned}$$

(101)

for all $k\in \mathbb {N}$ and $t\in [0,T]$.

Therefore, the estimate (101) yields subsequences of $({\tilde{u}}_k)_{k\in \mathbb {N}}$ and $({\tilde{\theta }}_k)_{k\in \mathbb {N}}$, that we still denote in the same way, and functions $({\tilde{u}},{\tilde{\theta }})$, such that

$$\begin{aligned}{} & {} {\tilde{u}}_k{\mathop {\rightharpoonup }\limits ^{*}} {\tilde{u}} \text{ in } L^{\infty }(0,T;H_0^1(\Omega )), \end{aligned}$$

(102)

$$\begin{aligned}{} & {} {\tilde{u}}_k'{\mathop {\rightharpoonup }\limits ^{*}} {\tilde{u}}' \text{ in } L^{\infty }(0,T;L^2(\Omega )), \end{aligned}$$

(103)

$$\begin{aligned}{} & {} {\tilde{\theta }}_k{\mathop {\rightharpoonup }\limits ^{*}} {\tilde{\theta }} \text{ in } L^{\infty }(0,T;L^2(\Omega )), \end{aligned}$$

(104)

as $k\rightarrow \infty$. Since $H_0^1(\Omega )$ is compactly embedded in $L^2(\Omega )$, from the Aubin-Lions Theorem, we have

$$\begin{aligned} {\tilde{u}}_k\rightarrow {\tilde{u}} \text{ in } L^{2}(0,T;L^2(\Omega )), \end{aligned}$$

(105)

as $k\rightarrow \infty$. From (97) and (104), we conclude that

$$\begin{aligned} {\tilde{\theta }}=0. \end{aligned}$$

(106)

For each $k\in \mathbb {N}$, $({\tilde{u}}_k,{\tilde{\theta }}_k)$ is a solution of

$$\begin{aligned}{} & {} \displaystyle {{\tilde{u}}_k''-c\Delta {\tilde{u}}_k+\text{ div }({\tilde{\theta }}_k)+\rho (x) \frac{g(u_k')}{c_k}=0} \,\, \text{ in }\,\Omega \times (0,T), \end{aligned}$$

(107)

$$\begin{aligned}{} & {} {\tilde{\theta }}_k'-\Delta {\tilde{\theta }}_k +\text{ div }({\tilde{u}}_k')=0\,\, \text{ in }\,\Omega \times (0,T), \end{aligned}$$

(108)

$$\begin{aligned}{} & {} {\tilde{u}}_k=0\,\, \text{ on }\,\Gamma \times (0,T), \end{aligned}$$

(109)

$$\begin{aligned}{} & {} {\tilde{\theta }}_k=0\,\, \text{ on }\,\Gamma _0\times (0,T), \end{aligned}$$

(110)

$$\begin{aligned}{} & {} {\tilde{\theta }}_k'-\beta \Delta _{\Gamma } {\tilde{\theta }}_k +\beta \frac{\partial {\tilde{\theta }}_k}{\partial \nu }-\alpha {\tilde{\theta }}_k=0\,\, \text{ on }\,\Gamma _1\times (0,T). \end{aligned}$$

(111)

Taking to the limit, as $k\rightarrow \infty$, and observing (95)–(97), and (102)–(106), we obtain

$$\begin{aligned}{} & {} \displaystyle {{\tilde{u}}''-c\Delta {\tilde{u}}=0} \,\, \text{ in }\,\Omega \times (0,T), \end{aligned}$$

(112)

$$\begin{aligned}{} & {} {\tilde{u}}=0\,\, \text{ on }\,\Gamma \times (0,T), \end{aligned}$$

(113)

$$\begin{aligned}{} & {} {\tilde{u}}'=0\,\, \text{ on }\,\omega \times (0,T). \end{aligned}$$

(114)

Thus, we can use the same arguments of the case $u\ne 0$ and to conclude that ${\tilde{u}}=0$. This and (106) give a contradiction with (93).

Therefore the claim (66) is proved. Combining (65) with (66), we obtain

$$\begin{aligned}&E(T) \le C \Big [ \int _0^T\int _{\Omega }|\nabla \theta |^2\,dx\,dt\nonumber \\&\quad +\int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma \,dt +\int _0^T\int _{\Omega }\rho (x)[u_t^2+g^2(u_t)]\,dx\,dt \Big ]. \end{aligned}$$

(115)

Define

$$\begin{aligned} \omega _A = \{ (x,t)\in \omega \times (0,T);\, |u_t(x,t)|>1 \} \end{aligned}$$

and

$$\begin{aligned} \omega _B=(\omega \times (0,T))\setminus \omega _A. \end{aligned}$$

Using Assumptions 1 and 2, we obtain

$$\begin{aligned} \int _{\omega _A}(u_t^2+g^2(u_t))\,dx\,dt \le \left( \frac{c_1^{-1}+c_2}{\rho _0} \right) \int _0^T\int _{\Omega } \rho (x)g(u_t)u_t\,dx\,dt. \end{aligned}$$

From (39), we have

$$\begin{aligned} \int _{\omega _B}(u_t^2+g^2(u_t))\,dx\,dt \le \int _{\omega _B} \varphi (g(u_t)u_t)\,dx\,dt. \end{aligned}$$

Using Jensen’s inequality, we obtain

$$\begin{aligned}{} & {} \int _{\omega _B}(u_t^2+g^2(u_t))\,dx\,dt\\{} & {} \quad \le \text{ meas }(\omega \times (0,T)) \varphi \left( \frac{1}{\text{ meas }(\omega \times (0,T))} \int _0^T\int _{\omega } \rho (x)g(u_t)u_t\,dx\,dt \right) \\{} & {} \quad \le \text{ meas }(\omega \times (0,T)) {\tilde{\varphi }}\left( \int _0^T\int _{\omega } \rho (x)g(u_t)u_t\,dx\,dt \right) . \end{aligned}$$

Thus

$$\begin{aligned}{} & {} \int _{\omega }(u_t^2+g^2(u_t))\,dx\,dt \le \left( \frac{c_1^{-1}+c_2}{\rho _0} \right) \int _0^T\int _{\Omega } \rho (x)g(u_t)u_t\,dx\,dt \\{} & {} \quad + \text{ meas }(\omega \times (0,T)) {\tilde{\varphi }}\left( \int _0^T\int _{\omega } \rho (x)g(u_t)u_t\,dx\,dt \right) . \end{aligned}$$

Since ${\tilde{\varphi }}$ is increasing and

$$\begin{aligned} \int _0^T\int _{\Omega }|\nabla \theta |^2\,dx\,dt +\int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma \,dt -\int _0^T\int _{\Gamma _1}\frac{\alpha }{\beta }\theta ^2\,dx\,dt \ge 0 \end{aligned}$$

we infer

$$\begin{aligned} \int _{\omega }(u_t^2+g^2(u_t))\,dx\,dt \le \left( \frac{c_1^{-1}+c_2}{\rho _0} \right) \Lambda + \text{ meas }(\omega \times (0,T)) {\tilde{\varphi }}\left( \Lambda \right) , \end{aligned}$$

(116)

where

$$\begin{aligned} \Lambda&= \int _0^T\int _{\Omega } \rho (x)g(u_t)u_t\,dx\,dt +\int _0^T\int _{\Omega }|\nabla \theta |^2\,dx\,dt\\&\quad +\int _0^T\int _{\Gamma _1}|\nabla _{\Gamma }\theta |^2\,d\Gamma \,dt -\int _0^T\int _{\Gamma _1}\frac{\alpha }{\beta }\theta ^2\,dx\,dt. \end{aligned}$$

Therefore, (115) and (116) give us that

$$\begin{aligned} E(T) \le C\left( \frac{c_1^{-1}+c_2}{\rho _0} \right) \Lambda + C\text{ meas }(\omega \times (0,T)) {\tilde{\varphi }}\left( \Lambda \right) . \end{aligned}$$

(117)

Since $L=\frac{1}{C\text{ meas }(\omega \times (0,T))}$ and $M=\frac{a_1^{-1}+a_2}{\rho _0\text{ meas }(\omega \times (0,T))}$, we have

$$\begin{aligned} E(T) \le \frac{M}{L}\Lambda +\frac{1}{L}{\tilde{\varphi }}(\Lambda ). \end{aligned}$$

Since p, defined in (41), is increasing, we obtain

$$\begin{aligned} p(E(T))\le \Lambda . \end{aligned}$$

This and Lemma 3.1 give us that

$$\begin{aligned} p(E(T)) +E(T)\le E(0). \end{aligned}$$

This inequality and Lemma 3.3 of Lasiecka and Tataru [31] give us the result. $\square$