A global nonexistence of solutions for a quasilinear viscoelastic wave equation with acoustic boundary conditions

Abstract

In this paper, we consider a quasilinear viscoelastic wave equation with acoustic boundary conditions. Under some appropriate assumption on the relaxation function g, the function Φ, $p > \max \{ \rho +2, m, q,2\}$, and the initial data, we prove a global nonexistence of solutions for a quasilinear viscoelastic wave equation with positive initial energy.

1 Introduction

In this paper, we are concerned with the following a quasi-nonlinear viscoelastic wave equation with acoustic boundary conditions:

$$\begin{aligned}& \begin{aligned}[b] & \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{tt} ( t )- \Delta u ( t )+ \int_{0}^{t} g ( t - s ) \Delta u ( s ) \,ds \\ &\quad {}+ \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t )= \bigl\vert u ( t ) \bigr\vert ^{p -2} u ( t ) \quad \mbox{in }\Omega \times(0,\infty), \end{aligned} \end{aligned}$$

(1)

$$\begin{aligned}& u ( t )=0\quad \mbox{on }\Gamma_{0} \times(0,\infty), \end{aligned}$$

(2)

$$\begin{aligned}& \frac{\partial u ( t )}{\partial \nu} - \int_{0}^{t} g ( t - s ) \frac{\partial u ( s )}{\partial \nu} \,ds + \Phi \bigl( u_{t} ( t )\bigr)= h ( x ) y_{t} ( t )\quad \mbox{on } \Gamma_{1} \times(0,\infty), \end{aligned}$$

(3)

$$\begin{aligned}& u_{t} ( t )+ f ( x ) y_{t} ( t )+ q ( x ) y ( t )=0\quad \mbox{on }\Gamma_{1} \times(0,\infty), \end{aligned}$$

(4)

$$\begin{aligned}& u ( x,0)= u_{0} ( x ), u_{t} ( x,0)= u_{1} ( x ) \quad \mbox{in }\Omega, \end{aligned}$$

(5)

$$\begin{aligned}& y ( x,0)= y_{0} ( x )\quad \mbox{on }\Gamma_{1}, \end{aligned}$$

(6)

where Ω is a regular and bounded domain of $\mathbb{R}^{n}$ ($n \geq1$), and $\partial \Omega = \Gamma_{0} \cup \Gamma_{1}$. Here $\Gamma_{0}$, $\Gamma_{1}$ are closed and disjoint and $\frac{\partial}{ \partial \nu}$ denotes the unit outer normal derivative to Γ. The function $g: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ is a positive nonincreasing function, the function $\Phi :\mathbb{R}\rightarrow \mathbb{R}$ is a monotone and continuous, and the functions $f, q, h: \Gamma_{1} \rightarrow \mathbb{R}_{+}$ are essentially bounded and $q ( x )\geq q_{0} >0$.

System (1)–(6) is a model of a quasilinear viscoelastic wave equation with acoustic boundary conditions. The acoustic boundary conditions were introduced by Morse and Ingard [14] in 1968 and developed by Beale and Rosencrans in [1], where the authors proved the global existence and regularity of the nonlinear problem. When $\vert u_{t} ( t ) \vert ^{\rho}$ is not a constant, system (1)–(6) can model materials whose density depends on the velocity $u_{t}$. The physical application of the above system is the problem of noise suppression in structural acoustic systems, which is one of great interests in physics and engineering. Also reducing the level of pressure in a helicopter’s cabin and suppressing the noise in the interior of an acoustic chamber are based on some special type of boundary conditions like those described in system (1)–(6), (see [4, 5] and another case [9]).

Boukhatem and Benabderrahmane [2, 3] studied the existence, blow-up, and decay of solutions for viscoelastic wave equations with acoustic boundary conditions. Recently, many authors have treated wave/beam equations with acoustic boundary conditions, see [7, 8, 10, 12, 13, 15, 16] and the references therein. Graber and Haid-Houari [5] studied the blow-up solutions for a nonlinear wave equation with porous acoustic boundary conditions:

$$\begin{aligned}& u_{tt} ( t )- \Delta u ( t )+ \alpha ( x ) u ( t )+ \phi ( u_{t} )= j_{1} \bigl( u ( t )\bigr)\quad \mbox{in }\Omega \times(0,\infty), \\& u ( t )=0\quad \mbox{on }\Gamma_{0} \times(0,\infty), \\& u_{t} ( t )+ f ( x ) z_{t} ( t )+ g ( x ) z ( t )=0\quad \mbox{on }\Gamma_{1} \times(0,\infty), \\& \frac{\partial u ( t )}{\partial \nu} - h ( x ) \eta \bigl( z_{t} ( t )\bigr)+ \rho \bigl( u_{t} ( t )\bigr)= j_{2} \bigl( u ( t )\bigr)\quad \mbox{on }\Gamma_{1} \times(0,\infty), \\& u ( x,0)= u_{0} ( x ), u_{t} ( x,0)= u_{1} ( x ) \quad \mbox{in }\Omega, \\& z ( x,0)= z_{0} ( x )\quad \mbox{on }\Gamma_{1}, \end{aligned}$$

where $\alpha: \Omega \rightarrow \mathbb{R}$ and $f, g, h: \overline{\Gamma}_{1} \rightarrow \mathbb{R}$ are given functions. Also the functions $j_{1}$ and $j_{2}$ are of a polynomial structure as follows: $j_{1} ( s )=\vert s \vert ^{p -2} s$, $j_{2} ( s )=\vert s \vert ^{k -2} s k$, $p \geq2$, the functions ρ and ϕ are monotone, continuous, and there exist four positive constants $m_{q}$, $M_{q}$, $c_{r}$, and $C_{r}$ such that $m_{q} \vert s \vert ^{q} \leq \rho ( s ) s \leq M_{q} \vert s \vert ^{q}$, $c_{r} \vert s \vert ^{r} \leq \phi ( s ) s \leq C_{r} \vert s \vert ^{r}$. In addition, Di et al. [4] studied a viscoelastic wave equation with nonlinear boundary source term:

$$\begin{aligned}& \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{tt} ( t )- \Delta u ( t )+ \int_{0}^{t} g ( t - s ) \Delta u ( s ) \,ds =0\quad \mbox{in }\Omega \times(0,\infty), \\& u ( x, t )=0\quad \mbox{on }\Gamma_{0} \times(0,\infty), \\& \frac{\partial u}{\partial \nu} ( t )- \int_{0}^{t} g ( t - s ) \frac{\partial u}{\partial \nu} ( s ) \,ds = f \bigl( u ( t )\bigr)\quad \mbox{on }\Gamma_{1} \times(0,\infty), \\& u ( x,0)= u_{0} ( x ), u_{t} ( x,0)= u_{1} ( x ) \quad \mbox{in }\Omega, \end{aligned}$$

where $\rho \geq1$ and Ω is a bounded domain of $\mathbb{R}^{n}$ ($n \geq1$) with smooth boundary $\Gamma:=\partial \Omega$. Let $\{ \Gamma_{0}, \Gamma_{1} \}$ be a partition of its boundary Γ such that $\Gamma = \Gamma_{0} \cup \Gamma_{1}$, $\overline{\Gamma}_{0} \cap \overline{\Gamma}_{1} =\emptyset$, and $\operatorname{meas}( \Gamma_{0} )>0$. Here, ν is the unit outward normal to Γ, and g, f are given functions satisfying suitable conditions. They introduced a family of potential wells and proved the invariance of some sets. Then they established the existence and nonexistence of a global weak solution with small initial energy under suitable assumptions on $g (\cdot)$, $f (\cdot)$, initial data, and the parameters in the equation. Also they showed the global existence of a weak solution for the problem with critical initial conditions $I ( u_{0} )\geq0$ and $e (0)= d$. Furthermore, Song [17] studied the nonlinear viscoelastic wave equation

$$\begin{aligned}& \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{tt} ( t )- \Delta u ( t )+ \int_{0}^{t} g ( t - \tau ) \Delta u ( \tau ) \,d \tau \\& \quad {}+ \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t )= \bigl\vert u ( t ) \bigr\vert ^{p -2} u ( t ) \quad \mbox{in }\Omega \times[0, T ], \\& u ( x, t )=0\quad \mbox{on }\partial \Omega \times[0, T ], \\& u ( x,0)= u_{0} ( x ), u_{t} ( x,0)= u_{1} ( x ) \quad \mbox{in }\Omega, \end{aligned}$$

where Ω is a bounded domain of $\mathbb{R}^{n}$ ($n \geq1$) with smooth boundary ∂Ω, $m \geq2$, $g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ a positive nonincreasing function, and

$$2< p, \rho < \infty,\quad \mbox{if }n =1,2,\qquad 2< p,\rho \leq \frac{2( n -1)}{n -2}\quad \mbox{if }n \geq3. $$

The author proved the global nonexistence of positive initial energy solutions for a viscoelastic wave equation. Recently Jeong et al. [6] investigated the quasilinear wave equation with acoustic boundary conditions

$$\begin{aligned}& u_{tt} ( t )- \Delta u_{t} ( t )- \operatorname{div} \bigl( \bigl\vert \nabla u ( t ) \bigr\vert ^{\alpha -2} \nabla u ( t )\bigr)- \operatorname{div} \bigl( \bigl\vert \nabla u_{t} ( t ) \bigr\vert ^{\beta -2} \nabla u_{t} ( t )\bigr)+ a \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t )\\& \quad = \bigl\vert u ( t ) \bigr\vert ^{p -2} u ( t )\quad \mbox{in }\Omega \times(0,\infty), \\& u ( t )=0\quad \mbox{on }\Gamma_{0} \times(0,\infty), \\& \frac{\partial u_{t} ( t )}{\partial \nu} + \bigl\vert \nabla u ( t ) \bigr\vert ^{\alpha -2} \frac{\partial u ( t )}{\partial \nu} + \bigl\vert \nabla u_{t} ( t ) \bigr\vert ^{\beta -2} \frac{\partial u_{t} ( t )}{\partial \nu} = h ( x ) y_{t} ( t )\quad \mbox{on } \Gamma_{1} \times(0,\infty), \\& u_{t} ( t ) + f ( x ) y_{t} ( t ) + q ( x ) y ( t ) =0\quad \mbox{on }\Gamma_{1} \times(0,\infty), \\& u ( x,0 ) = u_{0} ( x ),\qquad u_{t} ( x,0 ) = u_{1} ( x )\quad \mbox{in }\Omega, \\& y ( x,0)= y_{0} ( x )\quad \mbox{on }\Gamma_{1}, \end{aligned}$$

where $a, b >0$, $\alpha, \beta, m, p >2$, Ω is a regular and bounded domain of $\mathbb{R}^{n}$ ($n \geq1$) and $\partial \Omega (= \Gamma )= \Gamma_{0} \cup \Gamma_{1}$. The functions $f, q, h: \Gamma_{1} \rightarrow \mathbb{R}_{+}$ are essentially bounded. They studied the global nonexistence of solutions for a quasilinear wave equation with acoustic boundary conditions. Motivated by the previous works [5, 17], we consider problem (1)–(6). Under suitable assumptions on the relaxation function g, the nonlinear function $\Phi (\cdot)$, $p > \max \{ \rho +2, m, q,2\}$, the initial data, and the parameters in the system, we prove the nonexistence of a weak solution with small positive initial energy.

2 Blow-up result

In this section, we present some material which will be used throughout this work. First, we introduce the set

$$H_{\Gamma_{0}}^{1} ( \Omega )=\bigl\{ u \in H^{1} ( \Omega )\vert u =0 \mbox{ on }\Gamma_{0} \bigr\} , $$

and endow $H_{\Gamma_{0}}^{1} ( \Omega )$ with the Hilbert structure induced by $H^{1} ( \Omega )$. We have that $H_{\Gamma_{0}}^{1} ( \Omega )$ is a Hilbert space. For simplicity, we denote $\Vert \cdot\Vert _{p} =\Vert \cdot\Vert _{L^{p} ( \Omega )}$, $\Vert \cdot\Vert _{p, \Gamma} =\Vert \cdot\Vert _{L^{p} ( \Gamma )}$, $1\leq p \leq\infty$.

We present some assumptions and preliminaries needed in the proof of our main result.

We make the following assumptions:

(H1) :

$g: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+}$ is a differentiable function such that

$$ 1- \int_{0}^{\infty} g ( s ) \,ds = l >0,\qquad g ( t )\geq0, \qquad g '( t )\leq0, \quad \forall t \geq0. $$

(7)

(H2) :

For the nonlinear terms, we have

$$\begin{aligned}& 2< p \leq \frac{2( n -1)}{n -2} \quad \mbox{if }n \geq3\quad \mbox{and}\quad p >2 \quad \mbox{if }n =1,2, \end{aligned}$$

(8)

$$\begin{aligned}& 2< \rho \leq \frac{2}{n -2}\quad \mbox{if }n \geq3\quad \mbox{and}\quad \rho >0\quad \mbox{if }n =1,2. \end{aligned}$$

(9)

(H3) :

$\Phi :\mathbb{R}\rightarrow \mathbb{R}$ is monotone, continuous, and there exist positive constants $m_{q}$ and $M_{q}$ such that

$$ m_{q} \vert s \vert ^{q} \leq \Phi ( s ) s \leq M_{q} \vert s \vert ^{q},\quad \forall s \in \mathbb{R}. $$

(10)

(H4) :

The functions f, q, h are essentially bounded such that

$$f ( x )>0,\qquad q ( x )>0\quad \mbox{and}\quad h ( x )>0,\quad \forall x \in \Gamma_{1}. $$

We state, without a proof, a local existence which can be established by combining arguments of [4, 5].

Let assumptions (H1)–(H4) hold, $u_{0} \in H_{\Gamma_{0}}^{1} ( \Omega )$, $u_{1} \in L^{2} ( \Omega )$, and $y_{0} \in L^{2} ( \Gamma_{1} )$. Then problem (1)–(6) admits a weak local solution $( u, y )$ such that, for some $T >0$,

$$\begin{aligned}& u \in L^{\infty} \bigl([0, T ); H_{\Gamma_{0}}^{1} ( \Omega )\bigr), \\& u_{t} \in L^{\infty} \bigl([0, T ); L^{2} ( \Omega )\bigr)\cap L^{m} \bigl([0, T ); \Omega \bigr)\cap L^{q} \bigl([0, T ); \Gamma_{1} \bigr), \\& y \in L^{2} \bigl([0, T ); \Gamma_{1} \bigr). \end{aligned}$$

To obtain the global nonexistence result, we need the following lemmas.

Lemma 2.1

Assume that (H1)–(H4) hold. Let $u(t)$ be a solution of problem (1)–(6). Then the energy functional $E(t)$ of problem (1)–(6) is nonincreasing. Moreover, the following energy inequality holds:

(11)

where

(12)

and

(13)

Lemma 2.2

Suppose that (H1)–(H4) hold. Let $u(t)$ be the solution of problem (1)–(6). Furthermore, assume that

$$E (0)< E_{1} = \biggl( \frac{1}{2} - \frac{1}{p} \biggr) B_{1}^{- \frac{2 p}{p -2}} $$

and

$$\Vert \nabla u_{0} \Vert \geq B_{1}^{- \frac{p}{p -2}}, $$

where $B_{1} = B / l^{\frac{1}{2}}$ and B is the best constant of the Sobolev embedding $H_{0}^{1} ( \Omega ) \hookrightarrow L^{p} ( \Omega )$. Then there exists a constant $\beta > B_{1}^{- \frac{p}{p -2}}$ such that

$$ \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \geq \beta^{2},\quad \forall t >0 $$

(14)

and

$$ \bigl\Vert u ( t ) \bigr\Vert _{p} \geq B_{1} \beta,\quad \forall t >0. $$

(15)

Proof

From (6) and the embedding theorem, we have

(16)

where $\xi = ( ( 1- \int_{0}^{t} g ( s ) \,ds ) \Vert \nabla u ( t ) \Vert ^{2} )^{\frac{1}{2}}$. It is easy to see that $G ( \xi )$ takes its maximum for $\xi = \xi^{*} = B_{1}^{- \frac{p}{p -2}}$, which is strictly increasing for $0< \xi < \xi^{*}$, strictly decreasing for $\xi > \xi^{*}$, $G ( \xi )\rightarrow-\infty$ as $\xi \rightarrow\infty$, and

$$G \bigl( \xi^{*} \bigr)= \biggl( \frac{1}{2} - \frac{1}{p} \biggr) B_{1}^{- \frac{2 p}{p -2}} = E_{1}. $$

Since $E (0)< E_{1}$, there exists $\beta > \xi^{*}$ such that $G ( \beta )= E (0)$. Set $\xi_{0} =\Vert \nabla u (0)\Vert$, by (16), we see that

$$G ( \xi_{0} )\leq E (0)= G ( \beta ), $$

which implies that

$$\Vert \nabla u_{0} \Vert = \xi_{0} > \beta. $$

To prove (14), we suppose on the contrary that

$$\biggl( \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \biggr)^{\frac{1}{2}} < \beta $$

for some $t = t_{0} >0$. By the continuity of $( 1- \int_{0}^{t} g ( s ) \,ds ) \Vert \nabla u ( t ) \Vert ^{2}$, we may choose $t_{0}$ such that

$$\beta > \biggl( \biggl( 1- \int_{0}^{t_{0}} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t_{0} ) \bigr\Vert ^{2} \biggr)^{\frac{1}{2}} > \xi^{*}. $$

Then it follows from (16) that

$$E ( t_{0} )\geq G \biggl( \biggl( \biggl( 1- \int_{0}^{t_{0}} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t_{0} ) \bigr\Vert ^{2} \biggr)^{\frac{1}{2}} \biggr) > G ( \beta )= E (0), $$

which contradicts Lemma 2.1. Hence (14) is proved. Now we will prove (15). From (12), (13), (14), and Lemma 2.1, we deduce that

Thus the proof of Lemma 2.2 is complete. □

Theorem 2.1

Let $2< m< p$, $2\leq q< p$ and assume that (H1)–(H4) hold. Suppose that $\rho < p-2$, $0< \varepsilon_{0} < \frac{p}{2} -1$, and

$$ \int_{0}^{\infty} g ( s ) \,ds < \frac{\frac{p}{2} - C M_{q} \frac{\lambda^{q}}{q} -1- \varepsilon_{0}}{\frac{p}{2} -1- C M_{q} \frac{\lambda^{q}}{q} + \frac{1}{2 p}} $$

(17)

are satisfied, then there exists no global solution of problem (1)–(6) if

$$ E (0)< \biggl( 1- C M_{q} \frac{\lambda^{q}}{q} \frac{2}{p -2} - \frac{1}{ p ( p -2)} \frac{1- l}{l} \biggr) \biggl( \frac{1}{2} - \frac{1}{p} \biggr) B_{1}^{- \frac{2 p}{p -2}}, $$

(18)

and

$$ \Vert \nabla u_{0} \Vert > B_{1}^{- \frac{p}{p -2}}. $$

(19)

Proof

Assume that the solution $u ( t )$ of (1)–(6) is global. We set

$$ H ( t ) = E_{2} - E ( t ), $$

(20)

where the constant $E_{2} \in( E (0), E_{1} )$ shall be chosen later. By Lemma 2.1, the function $H ( t )$ is increasing. Then, for $t \geq s \geq0$,

(21)

Thus from (14) we get

$$ \begin{aligned}[b] H ( t )&\leq E_{2} - \frac{1}{2} \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} + \frac{1}{p} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\leq E_{1} - \frac{1}{2} \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} + \frac{1}{p} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\leq E_{1} - \frac{1}{2} B_{1}^{- \frac{2 p}{p -2}} + \frac{1}{p} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &= \biggl( \frac{1}{2} - \frac{1}{p} \biggr) B_{1}^{- \frac{2 p}{p -2}} - \frac{1}{2} B_{1}^{- \frac{2 p}{p -2}} + \frac{1}{p} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\leq \frac{1}{p} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p}. \end{aligned} $$

(22)

Now, we define

$$ \begin{aligned}[b] L ( t )&= H^{1- \sigma} ( t )+ \frac{\varepsilon}{\rho +1} \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{t} ( t ) u ( t ) \,dx \\ &\quad {}- \frac{\varepsilon}{2} \int_{\Gamma_{1}} h ( x ) f ( x ) y^{2} ( t ) \,d \Gamma - \varepsilon \int_{\Gamma_{1}} h ( x ) u ( t ) y ( t ) \,d \Gamma, \end{aligned} $$

(23)

where the constants $0< \sigma <1$, $\varepsilon >0$ shall be chosen later.

Taking a derivative of (23), using (7)–(10) and Lemma 2.1, we have

$$\begin{aligned} L '( t )&=(1- \sigma ) H^{- \sigma} ( t ) H '( t )+ \frac{\varepsilon}{\rho +1} \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} + \varepsilon \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{tt} ( t ) u ( t ) \,dx \\ &\quad {}- \varepsilon \int_{\Gamma_{1}} h ( x ) f ( x ) y ( t ) y_{t} ( t ) \,d \Gamma - \varepsilon \int_{\Gamma_{1}} h ( x ) u_{t} ( t ) y ( t ) \,d \Gamma \\ &\quad {}- \varepsilon \int_{\Gamma_{1}} h ( x ) u ( t ) y_{t} ( t ) \,d \Gamma \\ & =(1- \sigma ) H^{- \sigma} ( t ) H '( t )+ \frac{\varepsilon}{\rho +1} \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {}+ \varepsilon \int_{\Omega} u ( t ) \biggl[ \Delta u ( t )- \int_{0}^{t} g ( t - s ) \Delta u ( s ) \,ds - \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t )+ \bigl\vert u ( t ) \bigr\vert ^{p -2} u ( t ) \biggr] \,dx\hspace{-20pt} \\ &\quad {} - \varepsilon \int_{\Gamma_{1}} h ( x ) f ( x ) y ( t ) y_{t} ( t ) \,d \Gamma - \varepsilon \int_{\Gamma_{1}} h ( x ) u_{t} ( t ) y ( t ) \,d \Gamma \\ &\quad {}- \varepsilon \int_{\Gamma_{1}} h ( x ) u ( t ) y_{t} ( t ) \,d \Gamma \\ &=(1- \sigma ) H^{- \sigma} ( t ) H '( t )+ \frac{\varepsilon}{\rho +1} \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {}- \varepsilon \int_{\Omega} \bigl\vert \nabla u ( t ) \bigr\vert ^{2} \,dx + \varepsilon \int_{\Omega} \nabla u ( t ) \int_{0}^{t} g ( t - s )\nabla u ( s )\,ds\,dx \\ &\quad {}- \varepsilon \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t ) u ( t ) \,dx + \varepsilon \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\quad {}+ \varepsilon \int_{\Gamma_{1}} u ( t ) \biggl( \frac{\partial u ( t )}{\partial \nu} - \int_{0}^{t} g ( t - s ) \frac{\partial u ( s )}{\partial \nu} \,ds \biggr) \,d \Gamma \\ &\quad {}- \varepsilon \int_{\Gamma_{1}} h ( x ) y ( t ) \bigl( f ( x ) y_{t} ( t )+ u_{t} ( t )\bigr) \,d \Gamma \\ &\quad {}- \varepsilon \int_{\Gamma_{1}} h ( x ) u ( t ) y_{t} ( t ) \,d \Gamma \\ &=(1- \sigma ) H^{- \sigma} ( t ) H '( t )+ \frac{\varepsilon}{\rho +1} \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} - \varepsilon \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \\ &\quad {} + \varepsilon \int_{\Omega} \nabla u ( t ) \int_{0}^{t} g ( t - s )\nabla u ( s )\,ds\,dx - \varepsilon \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t ) u ( t ) \,dx \\ &\quad {}+ \varepsilon \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} - \varepsilon \int_{\Gamma_{1}} \Phi \bigl( u_{t} ( t )\bigr) u ( t ) \,d \Gamma + \varepsilon \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma. \end{aligned}$$

(24)

Exploiting Hölder’s and Young’s inequalities, for any $\varepsilon_{1}$ ($0< \varepsilon_{1} <1$), we obtain

(25)

Thus from (24) and (25), we arrive at

(26)

Consequently, from (11), (12), (20), and (26), we deduce that

(27)

From this relation and using

it follows that

$$\begin{aligned} L '( t )&\geq(1- \sigma ) H^{- \sigma} ( t ) \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} +(1- \sigma ) H^{- \sigma} ( t ) \int_{\Gamma_{1}} \Phi \bigl( u_{t} ( t )\bigr) u_{t} ( t ) \,d \Gamma \\ &\quad {}+ \varepsilon (1- \varepsilon_{1} ) pH ( t )- \varepsilon (1- \varepsilon_{1} ) p E_{2} \\ &\quad {}+ \varepsilon \biggl( \frac{1}{\rho +1} + \frac{p (1- \varepsilon_{1} )}{ \rho +2} \biggr) \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {}+ \varepsilon \varepsilon_{1} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} - \varepsilon \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t ) u ( t ) \,dx \\ &\quad {}- \varepsilon \int_{\Gamma_{1}} \Phi \bigl( u_{t} ( t )\bigr) u ( t ) \,d \Gamma + \varepsilon \biggl( \frac{(1- \varepsilon_{1} ) p}{2} +1 \biggr) \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \\ &\quad {}+ \varepsilon \biggl( \frac{(1- \varepsilon_{1} ) p}{2} -1 \biggr) \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \\ &\quad {}- \frac{\varepsilon}{2 p (1- \varepsilon_{1} )} \int_{0}^{t} g ( s ) \,ds \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \\ &\geq(1- \sigma ) H^{- \sigma} ( t ) \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} + pH ( t ) \varepsilon (1- \varepsilon_{1} ) \\ &\quad {}+ \varepsilon \biggl( \frac{1}{\rho +1} + \frac{p (1- \varepsilon_{1} )}{ \rho +2} \biggr) \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} \\ &\quad {}+ \varepsilon \biggl( \frac{(1- \varepsilon_{1} ) p}{2} -1 \biggr) \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \\ &\quad {}- \frac{\varepsilon}{2 p (1- \varepsilon_{1} )} \int_{0}^{t} g ( s ) \,ds \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} - \varepsilon (1- \varepsilon_{1} ) p E_{2} + \varepsilon \varepsilon_{1} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\quad {}- \varepsilon \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t ) u ( t ) \,d \Gamma - \varepsilon \int_{\Gamma_{1}} \Phi \bigl( u_{t} ( t )\bigr) u ( t ) \,d \Gamma \\ &\quad {}+(1- \sigma ) H^{- \sigma} ( t ) \int_{\Gamma_{1}} \Phi \bigl( u_{t} ( t )\bigr) u_{t} ( t ) \,d \Gamma \\ &\quad {}+ \varepsilon \biggl( \frac{(1- \varepsilon_{1} ) p}{2} +1 \biggr) \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma. \end{aligned}$$

(28)

From Hölder’s and Young’s inequalities, the condition $m < p$, (22), and the embedding theorem ($L^{p} ( \Omega ) \hookrightarrow L^{m} ( \Omega )$), we obtain

$$ \begin{aligned}[b] \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t ) u ( t ) \,d \Gamma &\leq \biggl( \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{m} \,dx \biggr)^{\frac{m -1}{m}} \biggl( \int_{\Omega} \bigl\vert u ( t ) \bigr\vert ^{m} \,dx \biggr)^{\frac{1}{m}} \\ &\leq \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m -1} \bigl\Vert u ( t ) \bigr\Vert _{m} \\ &\leq C \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m -1} \bigl\Vert u ( t ) \bigr\Vert _{p} \\ & \leq C \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m -1} \bigl\Vert u ( t ) \bigr\Vert _{p}^{1- \frac{p}{m}} \bigl\Vert u ( t ) \bigr\Vert _{p}^{\frac{p}{m}} \\ &\leq C \bigl\Vert u ( t ) \bigr\Vert _{p}^{1- \frac{p}{m}} \bigl( \varepsilon_{1} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + C ( \varepsilon_{1} ) \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} \bigr) \\ &\leq CH ( t )^{\frac{1}{p} - \frac{1}{m}} \bigl( \varepsilon_{2} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + C ( \varepsilon_{2} ) \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} \bigr), \end{aligned} $$

(29)

where C is a generic positive constant which might change from line to line and $\varepsilon_{2} > \varepsilon_{1} p^{1/ p -1/ m}$.

Here we choose

$$ 0< \sigma < \min \biggl( \frac{1}{\rho +2} - \frac{1}{p}, \frac{1}{m} - \frac{1}{p}, \frac{1}{2} - \frac{1}{p} \biggr) $$

(30)

and take $\alpha = \frac{m - p}{pm} + \sigma =- ( \frac{1}{m} - \frac{1}{p} ) + \sigma <0$. Then the properties (21) of the function $H ( t )$ show that

$$H ( t )^{\frac{1}{p} - \frac{1}{m}} = H ( t )^{- \sigma} H ( t )^{\alpha} \leq H ( t )^{- \sigma} H (0 )^{\alpha}. $$

Thus from inequality (30) it follows

$$ \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{m -2} u_{t} ( t ) u ( t ) \,d \Gamma \leq CH ( t )^{- \sigma} H (0 )^{\alpha} \bigl( \varepsilon_{2} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + C ( \varepsilon_{2} ) \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} \bigr). $$

(31)

Moreover, from (10), it is clear that

$$ \int_{\Gamma_{1}} \Phi \bigl( u_{t} ( t )\bigr) u_{t} ( t ) \,d \Gamma \geq m_{q} \bigl\Vert u_{t} ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} $$

(32)

and the following Young’s inequality

$$XY \leq \frac{\lambda^{\gamma} X^{\gamma}}{\gamma} + \frac{\lambda^{- \beta} Y^{\beta}}{\beta}, $$

$X, Y \geq0$, $\lambda >0$, $\gamma, \beta \in \mathbb{R}_{+}$ such that $\frac{1}{\gamma} + \frac{1}{\beta} =1$, then from (10), we get

$$ \begin{aligned}[b] \int_{\Gamma_{1}} \Phi \bigl( u_{t} ( t )\bigr) u ( t ) \,d \Gamma &\leq M_{q} \int_{\Gamma_{1}} \bigl\vert u_{t} ( t ) \bigr\vert ^{q -1} \bigl\vert u ( t ) \bigr\vert \,d \Gamma \\ &\leq M_{q} \frac{\lambda^{q}}{q} \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} + M_{q} \frac{q -1}{q} \lambda^{- \frac{q}{q -1}} \bigl\Vert u_{t} ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q}. \end{aligned} $$

(33)

Thus from (28) and (31)–(33), we deduce

$$ \begin{aligned}[b] L '( t )&\geq(1- \sigma ) H^{- \sigma} ( t ) \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} + \varepsilon \biggl[ \frac{1}{\rho +1} + \frac{p (1- \varepsilon_{1} )}{\rho +1} \biggr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} + \varepsilon (1- \varepsilon_{1} ) pH ( t )\hspace{-20pt} \\ &\quad {}+ \varepsilon \biggl[ \biggl( \frac{(1- \varepsilon_{1} ) p}{2} -1 \biggr) \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) - \frac{1}{ 2 p (1- \varepsilon_{1} )} \int_{0}^{t} g ( s ) \,ds \biggr] \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}\hspace{-20pt} \\ &\quad {}- \varepsilon (1- \varepsilon_{1} ) p E_{2} + \varepsilon \varepsilon_{1} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\quad {}- \varepsilon CH ( t )^{- \sigma} H (0 )^{\alpha} \bigl( \varepsilon_{2} \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + C ( \varepsilon_{2} ) \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} \bigr) \\ &\quad {}+(1- \sigma ) H^{- \sigma} ( t ) m_{q} \bigl\Vert u_{t} ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \\ &\quad {}- \varepsilon M_{q} \frac{\lambda^{q}}{q} \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} - \varepsilon M_{q} \frac{q -1}{q} \lambda^{- \frac{q}{q -1}} \bigl\Vert u_{t} ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \\ &\quad {}+ \varepsilon \biggl( \frac{(1- \varepsilon_{1} ) p}{2} +1 \biggr) \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \\ &= H^{- \sigma} ( t )\bigl[1- \sigma - \varepsilon C H^{\alpha} (0) C ( \varepsilon_{2} )\bigr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} + \varepsilon \biggl[ \frac{1}{\rho +1} + \frac{p (1- \varepsilon_{1} )}{\rho +2} \biggr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2}\hspace{-20pt} \\ &\quad {}+ \varepsilon (1- \varepsilon_{1} ) pH ( t )- \varepsilon (1- \varepsilon_{1} ) p E_{2} \\ &\quad {}+ \varepsilon \biggl[ \biggl( \frac{(1- \varepsilon_{1} ) p}{2} -1 \biggr) \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) - \frac{1}{ 2 p (1- \varepsilon_{1} )} \int_{0}^{t} g ( s ) \,ds \biggr] \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}\hspace{-20pt} \\ &\quad {}+ \varepsilon \bigl[ \varepsilon_{1} - \varepsilon_{2} C H^{- \sigma} ( t ) H^{\alpha} (0)\bigr] \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} - \varepsilon M_{q} \frac{\lambda^{q}}{q} \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \\ &\quad {}+\biggl[(1- \sigma ) H^{- \sigma} ( t ) m_{q} - \varepsilon M_{q} \frac{q -1}{q} \lambda^{- \frac{q}{q -1}} \biggr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \\ &\quad {}+ \varepsilon \biggl( \frac{(1- \varepsilon_{1} ) p}{2} +1 \biggr) \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma,\quad \forall t \geq T_{0}. \end{aligned} $$

(34)

We also use the embedding theorem. Let us recall the inequality (C denotes a generic positive constant)

$$\bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}} \leq C \bigl\Vert u ( t ) \bigr\Vert _{H^{s} ( \Omega )}, $$

where $q \geq1$ and $0\leq s <1$, $s \geq \frac{N}{2} - \frac{N -1}{q} >0$ and the interpolation and Poincaré’s inequality (see [11])

$$\bigl\Vert u ( t ) \bigr\Vert _{H^{s} ( \Omega )} \leq C \bigl\Vert u ( t ) \bigr\Vert ^{1- s} \bigl\Vert \nabla u ( t ) \bigr\Vert ^{s} \leq C \bigl\Vert u ( t ) \bigr\Vert _{p}^{1- s} \bigl\Vert \nabla u ( t ) \bigr\Vert ^{s}. $$

If $s < \frac{2}{q}$, using again Young’s inequality, we obtain

$$\bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \leq C \bigl[ \bigl( \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \bigr)^{\frac{q (1- s ) \mu}{p}} +\bigl( \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \bigr)^{\frac{qs \theta}{2}} \bigr] $$

for $\frac{1}{\mu} + \frac{1}{\theta} =1$. Here we choose $\theta = \frac{2}{qs}$ to get $\mu = \frac{2}{2- qs}$. Therefore the previous inequality

$$ \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \leq C \biggl[ \bigl( \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \bigr)^{\frac{2 q (1- s )}{(2- qs ) p}} + \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \biggr]. $$

(35)

Now, choosing s such that

$$0< s \leq \frac{2( p - q )}{q ( p -2)}, $$

we get

$$ \frac{2 q (1- s )}{(2- qs ) p} \leq1. $$

(36)

Once inequality (36) is satisfied, we use the classical algebraic inequality

$$ \chi^{\nu} \leq( \chi +1)\leq \biggl( 1+ \frac{1}{w} \biggr) ( \chi + w ),\quad \forall \chi \geq0,0< \nu \leq1, w \geq0, $$

(37)

with $\chi =\Vert u ( t ) \Vert _{p}^{p}$, $d =1+ \frac{1}{H (0)}$, $w = H (0)$, and $\nu = \frac{2 q (1- s )}{(2- qs ) p}$ to get the following estimate:

$$ \bigl( \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \bigr)^{\frac{2 q (1- s )}{(2- qs ) p}} \leq d \bigl( \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + H (0)\bigr)\leq d \bigl( \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + H ( t )\bigr),\quad \forall t \geq0. $$

(38)

From (35) and (38), we have

$$ \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \leq C \biggl( \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} +2 H ( t )\biggr). $$

(39)

Inserting estimate (39) into (34) and using (14), we arrive at

$$\begin{aligned} L '( t )&\geq H^{- \sigma} ( t )\bigl[1- \sigma - \varepsilon C H^{\alpha} (0) C ( \varepsilon_{2} )\bigr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} \\ &\quad {}+ \varepsilon \biggl[ \frac{1}{\rho +1} + \frac{(1- \varepsilon_{1} ) p}{ \rho +2} \biggr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} + \varepsilon (1- \varepsilon_{1} ) pH ( t ) \\ &\quad {}+ \varepsilon \biggl[ \biggl( \frac{p (1- \varepsilon_{1} )}{2} -1 \biggr) \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) - \frac{1}{ 2 p (1- \varepsilon_{1} )} \int_{0}^{t} g ( s ) \,ds \biggr] \\ &\quad {}\times \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} - \varepsilon (1- \varepsilon_{1} ) p E_{2} + \varepsilon \bigl[ \varepsilon_{1} - \varepsilon_{2} C H^{- \sigma} ( t ) H^{\alpha} (0) \bigr] \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\quad {}- \varepsilon C M_{q} \frac{\lambda^{q}}{q} \biggl[ \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} +2 H ( t )\biggr] \\ &\quad {}+ \biggl[ (1- \sigma ) H^{- \sigma} ( t ) m_{q} - \varepsilon M_{q} \frac{q -1}{1} \lambda^{\frac{q}{q -1}} \biggr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \\ &\quad {}+ \varepsilon \biggl[ \frac{(1- \varepsilon_{1} ) p}{2} +1 \biggr] \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \\ &= H^{- \sigma} ( t )\bigl[1- \sigma - \varepsilon C H^{\alpha} (0) C ( \varepsilon_{2} )\bigr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} + \varepsilon \biggl[ \frac{1}{\rho +1} + \frac{(1- \varepsilon_{1} ) p}{\rho +2} \biggr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2}\hspace{-20pt} \\ &\quad {}+ \varepsilon \biggl[ (1- \varepsilon_{1} ) p -2 C M_{q} \frac{\lambda^{q}}{ q} \biggr] H ( t )- \varepsilon (1- \varepsilon_{1} ) p E_{2} \\ &\quad {}+ \varepsilon \biggl[ \biggl( \frac{(1- \varepsilon_{1} ) p}{2} - C M_{q} \frac{\lambda^{q}}{q} -1 \biggr) \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \\ &\quad {} - \frac{1}{2(1- \varepsilon_{1} ) p} \int_{0}^{t} g ( s ) \,ds \biggr] \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} \\ &\quad {}+ \varepsilon \biggl[ \varepsilon_{1} - \varepsilon_{2} C H^{- \sigma} ( t ) H^{\alpha} (0)- C M_{q} \frac{\lambda^{q}}{q} \biggr] \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\quad {}+ \biggl[ (1- \sigma ) H^{- \sigma} ( t ) m_{q} - \varepsilon M_{q} \frac{q -1}{q} \lambda^{- \frac{q}{q -1}} \biggr] \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \\ &\quad {}+ \varepsilon \biggl[ \frac{(1- \varepsilon_{1} ) p}{2} +1 \biggr] \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \\ &\geq H^{- \sigma} ( t ) \bigl[ 1- \sigma - \varepsilon C H^{\alpha} (0) C ( \varepsilon_{2} ) \bigr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} + \varepsilon \biggl[ \frac{1}{\rho +1} + \frac{(1- \varepsilon_{1} ) p}{ \rho +2} \biggr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2}\hspace{-20pt} \\ &\quad {}+ \varepsilon \biggl[ (1- \varepsilon_{1} ) p -2 C M_{q} \frac{\lambda^{q}}{ q} \biggr] H ( t )- \varepsilon (1- \varepsilon_{1} ) p E_{2} \\ &\quad {}+ \frac{\varepsilon}{1- \int_{0}^{t} g ( s ) \,ds} \biggl[ \biggl( \frac{(1- \varepsilon_{1} ) p}{2} - C M_{q} \frac{\lambda^{q}}{q} -1 \biggr) l - \frac{1}{2(1- \varepsilon_{1} ) p} (1- l ) \biggr] \beta^{2} \\ &\quad {}+ \varepsilon \biggl[ \varepsilon_{1} - \varepsilon_{2} C H^{- \sigma} ( t ) H^{\alpha} (0)- C M_{q} \frac{\lambda^{q}}{q} \biggr] \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\quad {}+ \biggl[ (1- \sigma ) H^{- \sigma} ( t ) m_{q} - \varepsilon M_{q} \frac{q -1}{q} \lambda^{- \frac{q}{q -1}} \biggr] \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \\ &\quad {}+ \varepsilon \biggl[ \frac{(1- \varepsilon_{1} ) p}{2} +1 \biggr] \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma. \end{aligned}$$

(40)

Since

$$\int_{0}^{\infty} g ( s ) \,ds < \frac{\frac{p}{2} -1- C M_{q} \frac{\lambda^{q}}{q} - \varepsilon_{0}}{\frac{p}{2} -1- C M_{q} \frac{\lambda^{q}}{q} + \frac{1}{2 p}},\quad p >2, $$

we have

$$\biggl( \frac{p}{2} -1- C M_{q} \frac{\lambda^{q}}{q} \biggr) \biggl( 1- \int_{0}^{\infty} g ( s ) \,ds \biggr) - \frac{1}{2 p} \int_{0}^{\infty} g ( s ) \,ds > \varepsilon_{0} >0. $$

It is easy to see that there exist $\varepsilon_{1}^{*} >0$ and $T_{0} >0$ such that, for $0< \varepsilon_{1} < \varepsilon_{1}^{*}:=1- \frac{2 ( 1+ \varepsilon_{0} )}{p}$, $0< \varepsilon_{0} < \frac{p}{2} -1$, and $t > T_{0}$,

$$\begin{gathered} \frac{ [ ( \frac{(1- \varepsilon_{1} ) p}{2} - C M_{q} \frac{\lambda^{q}}{q} -1 ) l - \frac{1}{2(1- \varepsilon_{1} ) p} (1- l ) ] \beta^{2}}{1- \int_{0}^{t} g ( s ) \,ds} \\ \quad > \frac{ ( \frac{(1- \varepsilon_{1} ) p}{2} - C M_{q} \frac{\lambda^{q}}{q} -1 ) l - \frac{1}{2(1- \varepsilon_{1} ) p} (1- l )}{1- \int_{0}^{\infty} g ( s ) \,ds} B_{1}^{- \frac{2 p}{p -2}}. \end{gathered} $$

Now, we may choose $\varepsilon_{1} >0$ sufficiently small and $E_{2} \in( E (0), E_{1} )$ sufficiently near $E (0)$ such that

$$ \frac{ ( \frac{(1- \varepsilon_{1} ) p}{2} - C M_{q} \frac{\lambda^{q}}{q} -1 ) l - \frac{1}{2(1- \varepsilon_{1} ) p} (1- l )}{1- \int_{0}^{\infty} g ( s ) \,ds} B_{1}^{- \frac{2 p}{p -2}} - p (1- \varepsilon_{1} ) E_{2} >0, $$

(41)

since

$$\begin{aligned} E (0)&< E_{2} < \biggl( \frac{1}{2} - C M_{q} \frac{\lambda^{q}}{q} \frac{1}{p} - \frac{1}{p} - \frac{1}{2 p^{2}} \frac{1- l}{l} \biggr) B_{1}^{- \frac{2 p}{p -2}} \\ &= \biggl( 1- C M_{q} \frac{\lambda^{q}}{q} \frac{2}{p -2} - \frac{1}{p ( p -2)} \frac{1- l}{l} \biggr) \biggl( \frac{1}{2} - \frac{1}{p} \biggr) B_{1}^{- \frac{2 p}{p -2}} < \biggl( \frac{1}{2} - \frac{1}{p} \biggr) B_{1}^{- \frac{2 p}{p -2}}. \end{aligned} $$

From (40) and (41), we arrive at

$$ \begin{aligned}[b] L '( t )&\geq H^{- \sigma} ( t ) \bigl[ 1- \sigma - \varepsilon C H^{\alpha} (0) C ( \varepsilon_{2} ) \bigr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{m}^{m} + \varepsilon \biggl[ \frac{1}{\rho +1} + \frac{(1- \varepsilon_{1} ) p}{ \rho +2} \biggr] \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2}\hspace{-20pt} \\ &\quad {}+ \varepsilon \biggl[ (1- \varepsilon_{1} ) p -2 C M_{q} \frac{\lambda^{q}}{ q} \biggr] H ( t ) \\ &\quad {}+ \varepsilon \biggl[ \varepsilon_{1} - \varepsilon_{2} C H^{- \sigma} ( t ) H^{\alpha} (0)- C M_{q} \frac{\lambda^{q}}{q} \biggr] \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \\ &\quad {}+ \biggl[ (1- \sigma ) H^{- \sigma} ( t ) m_{q} - \varepsilon M_{q} \frac{q -1}{q} \lambda^{- \frac{q}{q -1}} \biggr] \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} \\ &\quad {}+ \varepsilon \biggl[ \frac{ ( 1- \varepsilon_{1} ) p}{2} +1 \biggr] \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma. \end{aligned} $$

(42)

At this point, for $\varepsilon_{2} C H^{- \sigma} ( t ) H^{\alpha} (0)< \varepsilon_{1} <\min \{1, \varepsilon_{2} p^{1/ m -1/ p} \}$, we may take λ sufficiently small such that

$$\begin{gathered} (1- \varepsilon_{1} ) p -2 C M_{q} \frac{\lambda^{q}}{q} >0, \\ \varepsilon_{1} - \varepsilon_{2} C H^{- \sigma} ( t ) H^{\alpha} (0)- C M_{q} \frac{\lambda^{q}}{q} >0. \end{gathered} $$

Once again, we choose ε small enough such that

$$\begin{gathered} 1- \sigma - \varepsilon C H^{\alpha} (0) C ( \varepsilon_{2} )>0, \\ (1- \sigma ) H^{- \sigma} ( t ) m_{q} - \varepsilon M_{q} \frac{q -1}{q} \lambda^{- \frac{q}{q -1}} >0. \end{gathered} $$

Then from (42) there exists a positive constant $K_{1} >0$ such that following inequality holds:

$$ L '( t )\geq \varepsilon K_{1} \biggl( H ( t )+ \bigl\Vert u ( t ) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} + \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr). $$

(43)

On the other hand, from definition (23) and since $f, h >0$, we have

$$L ( t )\leq H^{1- \sigma} ( t )+ \frac{\varepsilon}{\rho +1} \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{t} ( t ) u ( t ) \,dx K_{1} - \varepsilon \int_{\Gamma_{1}} h ( x ) q ( x ) y ( t ) \,d \Gamma. $$

Consequently, the above estimate leads to

$$ \begin{aligned}[b] L^{\frac{1}{1- \sigma}} ( t ) &\leq C ( \varepsilon, \sigma, \rho ) \biggl[ H ( t )+ \biggl( \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{t} ( t ) u ( t ) \,dx \biggr)^{\frac{1}{1- \sigma}} \\ &\quad {}+ \biggl( \int_{\Gamma_{1}} h ( x ) q ( x ) y ( t ) \,d \Gamma \biggr)^{\frac{1}{1- \sigma}} \biggr]. \end{aligned} $$

(44)

We now estimate (see [17])

$$\begin{aligned} \biggl( \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{t} ( t ) u ( t ) \,dx \biggr)^{\frac{1}{1- \sigma}} &\leq C \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\frac{\rho +1}{1- \sigma}} \bigl\Vert u ( t ) \bigr\Vert _{p}^{\frac{1}{1- \sigma}} \\ &\leq C \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\frac{\rho +1}{1- \sigma} \mu} \bigl\Vert u ( t ) \bigr\Vert _{p}^{\frac{1}{1- \sigma} \theta}, \end{aligned} $$

where $\frac{1}{\mu} + \frac{1}{\theta} =1$. Choose $\mu = \frac{(1- \sigma )( \rho +2)}{\rho +1} >1$, then

$$\frac{\theta}{1- \sigma} = \frac{\rho +2}{(1- \sigma )( \rho +2)-( \rho +1)}. $$

From (30), we know

$$ \frac{\theta}{1- \sigma} < p. $$

(45)

Then from (22) we deduce

$$ \begin{aligned}[b] \bigl\Vert u ( t ) \bigr\Vert _{p}^{\frac{\theta}{1- \sigma}} &= \bigl\Vert u ( t ) \bigr\Vert _{p}^{p - ( p - \frac{\theta}{1- \sigma} )} = \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} \bigl\Vert u ( t ) \bigr\Vert _{p}^{- k} \\ &\leq C \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} H (0 )^{- \frac{k}{p}}, \end{aligned} $$

(46)

where $k = p - \frac{\theta}{1- \sigma}$ is a positive constant. Thus from (46) we obtain

$$ \biggl( \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{t} ( t ) u ( t ) \,dx \biggr)^{\frac{1}{1- \sigma}} \leq C \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} H (0 )^{- \frac{k}{p}}. $$

(47)

On the other hand, by the same method as in [13], we obtain

$$\begin{aligned} \biggl\vert \int_{\Gamma_{1}} h ( x ) u ( t ) y ( t ) \,d \Gamma \biggr\vert &= \biggl\vert \int_{\Gamma_{1}} \frac{h ( x ) q ( x )}{q ( x )} u ( t ) y ( t ) \,d \Gamma \biggr\vert \\ &\leq \frac{ \Vert h \Vert _{L^{\infty}}^{\frac{1}{2}} \Vert q \Vert _{L^{\infty}}^{\frac{1}{2}}}{q_{0}} \biggl( \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr)^{\frac{1}{2}} \biggl( \int_{\Gamma_{1}} u^{2} ( t ) \,d \Gamma \biggr)^{\frac{1}{2}}. \end{aligned} $$

Using the embedding $L^{p} ( \Gamma_{1} ) \hookrightarrow L^{2} ( \Gamma_{1} )$ and Young’s inequality, we get

$$\begin{aligned} \biggl\vert \int_{\Gamma_{1}} h ( x ) u ( t ) y ( t ) \,d \Gamma \biggr\vert &= \biggl\vert \int_{\Gamma_{1}} \frac{h ( x ) q ( x )}{q ( x )} u ( t ) y ( t ) \,d \Gamma \biggr\vert \\ &\leq \tilde{C} \frac{ \Vert h \Vert _{L^{\infty}}^{\frac{1}{2}} \Vert q \Vert _{L^{\infty}}^{\frac{1}{2}}}{q_{0}} \biggl( \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr)^{\frac{1}{2}} \biggl( \int_{\Gamma_{1}} u^{q} ( t ) \,d \Gamma \biggr)^{\frac{1}{q}}. \end{aligned} $$

Consequently, there exists a positive constant $\tilde{C}_{1} = \tilde{C}_{1} (\Vert h \Vert _{L^{\infty}},\Vert q \Vert _{L^{\infty}}, q_{0}, \sigma )$ such that

$$\begin{aligned} & \biggl( \int_{\Gamma_{1}} h ( x ) u ( t ) y ( t ) \,d \Gamma \biggr)^{\frac{1}{1- \sigma}} \\ &\quad \leq \tilde{C}_{1} \biggl( \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr)^{\frac{1}{2(1- \sigma )}} \biggl( \int_{\Gamma_{1}} u^{q} ( t ) \,d \Gamma \biggr)^{\frac{1}{q (1- \sigma )}}. \end{aligned} $$

Applying Young’s inequality to the right-hand side of the preceding inequality, there exists a positive constant, also denoted by $\tilde{C}_{2}$, such that

$$ \begin{aligned}[b] & \biggl( \int_{\Gamma_{1}} h ( x ) u ( t ) y ( t ) \,d \Gamma \biggr)^{\frac{1}{1- \sigma}} \\ &\quad \leq \tilde{C}_{2} \biggl( \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr)^{\frac{\theta}{2(1- \sigma )}} \biggl( \int_{\Gamma_{1}} \bigl\vert u ( t ) \bigr\vert ^{q} \,d \Gamma \biggr)^{\frac{\tau}{q (1- \sigma )}} \end{aligned} $$

(48)

for $\frac{1}{\tau} + \frac{1}{\theta} =1$. We take $\theta =2(1- \sigma )$, hence $\tau = \frac{2(1- \sigma )}{1-2 \sigma}$ to get

$$ \begin{aligned}[b] & \biggl( \int_{\Gamma_{1}} h ( x ) u ( t ) y ( t ) \,d \Gamma \biggr)^{\frac{1}{1- \sigma}} \\ &\quad \leq C \biggl[ \biggl( \int_{\Gamma_{1}} \bigl\vert u ( t ) \bigr\vert ^{q} \,d \Gamma \biggr)^{\frac{2}{q (1- \sigma )}} + \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr].\end{aligned} $$

(49)

By using (30) and the algebraic inequality (37) with $\chi = \int_{\Gamma_{1}} \vert u ( t ) \vert ^{p} \,d \Gamma$, $d =1+ \frac{1}{H (0)}$, $w = H (0)$, and $\nu = \frac{2}{p (1-2 \sigma )}$, condition (30) on σ ensures that $0< \nu <1$, and we get

$$\chi^{\nu} \leq d \bigl( \chi + H (0)\bigr)\leq d \bigl( \chi + H ( t ) \bigr). $$

Therefore from (49) there exists a positive constant C̃ such that, for all $t \geq0$,

$$ \begin{aligned}[b] & \biggl( \int_{\Gamma_{1}} h ( x ) u ( t ) y ( t ) \,d \Gamma \biggr)^{\frac{1}{1- \sigma}} \\ &\quad \leq \tilde{C} \biggl( H ( t )+ \bigl\Vert u ( t ) \bigr\Vert _{q, \Gamma_{1}}^{q} + \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr). \end{aligned} $$

(50)

Thus from (39) and (50) we get

$$ \begin{aligned}[b] & \biggl( \int_{\Gamma_{1}} h ( x ) u ( t ) y ( t ) \,d \Gamma \biggr)^{\frac{1}{1- \sigma}} \\ &\quad \leq C \biggl( H ( t )+ \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + \biggl( 1- \int_{0}^{t} g ( s ) \,ds \biggr) \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} + \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr) \\ &\quad \leq C \biggl( H ( t )+ \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} + \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr), \end{aligned} $$

(51)

where C is a positive constant. Therefore, from (44), (47), and (51), we arrive at

$$ \begin{aligned}[b] L^{\frac{1}{1- \sigma}} ( t )&\leq \overline{C} \biggl[ H ( t )+ \biggl( \int_{\Omega} \bigl\vert u_{t} ( t ) \bigr\vert ^{\rho} u_{t} ( t ) u ( t ) \,dx \biggr)^{\frac{1}{1- \sigma}} + \biggl( \int_{\Gamma_{1}} h ( x ) q ( x ) y ( t ) \,d \Gamma \biggr)^{\frac{1}{1- \sigma}} \biggr] \\ &\leq \overline{C} \biggl[ H ( t )+ \bigl\Vert u_{t} ( t ) \bigr\Vert _{\rho +2}^{\rho +2} + \bigl\Vert u ( t ) \bigr\Vert _{p}^{p} + \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2} + \int_{\Gamma_{1}} h ( x ) q ( x ) y^{2} ( t ) \,d \Gamma \biggr], \end{aligned} $$

(52)

where C̅ is a constant depending on ε, σ, ρ, C̃, C. Consequently, combining (43) and (52), for some $\xi >0$, we get

$$L' ( t )\geq \xi L^{\frac{1}{1- \sigma}} ( t ), \quad \forall t \geq0. $$

For ε sufficiently small, there exists some constant $T_{1}$ such that

$$\begin{aligned} L ( T_{1} )&= H^{1- \sigma} ( T_{1} )+ \frac{\varepsilon}{\rho +1} \int_{\Omega} \bigl\vert u_{t} ( T_{1} ) \bigr\vert ^{\rho} u_{t} ( T_{1} ) u ( T_{1} ) \,dx \\ &\quad {}- \frac{\varepsilon}{2} \int_{\Gamma_{1}} h ( x ) f ( x ) y^{2} ( T_{1} ) \,d \Gamma - \varepsilon \int_{\Gamma_{1}} h ( x ) u ( t ) y ( T_{1} ) \,d \Gamma\\ & >0. \end{aligned} $$

Hence we get

$$ L' ( t )\geq \xi L^{\frac{1}{1- \sigma}} ( t )>0,\quad \forall t \geq T_{1}. $$

(53)

A simple integration of (53) over $( T_{1}, t)$ yields

$$L^{\frac{\sigma}{1- \sigma}} ( t )\geq \frac{1}{L^{\frac{- \sigma}{1- \sigma}} ( T_{1} )- \frac{\xi\sigma}{1- \sigma} t}, \quad \forall t \geq T_{1}. $$

Hence $L ( t )$ blows up in finite time

$$T^{*} \leq \frac{1- \sigma}{\xi\sigma L^{\frac{\sigma}{1- \sigma}} ( T_{1} )}. $$

Thus the proof of Theorem 2.1 is complete. □

3 Conclusion

In this paper, we consider a quasilinear viscoelastic wave equation with acoustic boundary condition. Under some appropriate assumption on the relaxation function g, the functionΦ, $p > \max \{ \rho +2, m, q,2 \}$, and the initial data, we prove a global nonexistence of solutions for a quasilinear viscoelastic wave equation with positive initial energy. Actually, the principle result of the paper, Theorem 2.1, is a global nonexistence result in the case where the interior source term $\vert u \vert ^{p -2} u$ dominates both the interior and boundary damping terms, $\vert u_{t} \vert ^{m -2} u_{t}$ and $\Phi ( u_{t} )\sim\vert u_{t} \vert ^{q -2} u_{t}$, in an appropriate sense under the added assumption that the initial total energy is sufficiently small.

4 Abbreviations

A quasilinear viscoelastic wave equation with acoustic boundary condition is considered; Some assumptions and needed lemmas are presented; The nonexistence of the weak solution with small positive initial energy is proved by suitable assumptions on the relaxation function g, the nonlinear function $\Phi (\cdot)$, $p > \max \{ \rho +2, m, q,2\}$, the initial data, and the parameters in the system.