Abstract
In this paper, we are concerned with the energy decay rate of the nonlinear viscoelastic problem with dynamic and acoustic boundary conditions.
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1 Introduction
In this paper, we are concerned with the energy decay rate of the following nonlinear viscoelastic problem with a time-varying delay in the boundary feedback and acoustic boundary conditions:
where Ω regular and is a bounded domain of \(R^{n} \), \(n\geq1 \), \(\partial\Omega=\Gamma_{0} \cup\Gamma_{1} \). Here \(\Gamma_{0} \), \(\Gamma_{1} \) are closed and disjoint with \(\operatorname{meas} (\Gamma_{0} )>0\) and ν is the unit outward normal to ∂Ω, \(\delta_{0} >0\), \(\delta_{1} \geq0\), \(m\geq2\), \(p>2\), g denotes the memory kernel and a, b are real valued functions which satisfy appropriate conditions. The functions \(f,m,h: \Gamma_{1} \to R^{+}\) are essentially bounded, \(k_{1}, k_{2} : R\to R\) are given functions, \(\tau(t)>0\) represents the time-varying delay, \(\mu_{1}\), \(\mu_{2} \) are real numbers with \(\mu_{1} >0\), \(\mu_{2} \neq0\) and the initial data \((u_{0}, u_{1}, y_{0} )\) belongs to a suitable space. This type of equation usually arises in the theory of viscoelasticity. It is well known that viscoelastic materials have memory effects, which is due to the mechanical response influenced by the history of the materials themselves. From the mathematical point of view, their memory effects are modeled by integrodifferential equations. Hence, equations related to the behavior of the solutions for the PDE system have attracted considerable attention in recent years. We can refer to recent work in [1–7].
The dynamic boundary conditions are not only important from the theoretical point of view but also arise in numerous practical problems. Among the early results dealing with this type of boundary conditions are those of [8, 9] in which the author has made contributions to this field. Recently, some authors have studied the existence and decay of solutions for a wave equation with dynamic boundary conditions [10–13].
Moreover, the acoustic boundary conditions was introduced by Morse and Ingard in [14] and developed by Beale and Rosencrans in [15], where the authors proved the global existence and regularity of the linear problem. Recently, some authors have studied the existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions (see [16–19]). Time delay so often arises in many physical, chemical, biological and economical phenomena because these phenomena depend not only on the present state but also on the history of the system in a more complicated way.
In recent years, differential equations with time delay effects have become an active area of research; see for example [20–26] and the references therein. To stabilize a hyperbolic system involving input delay terms, additional control terms will be necessary. For instance in [22], the authors are proved the boundary stabilization of a nonlinear viscoelastic equation with interior time-varying delay and nonlinear dissipative boundary feedback. In particular, Wu and Chen [27] consider the nonlinear viscoelastic wave equation with boundary dissipation
where \(K_{0} >0\) and Ω is a bounded domain in \(R^{n}\) (\(n\geq1\)) with a smooth boundary \(\Gamma=\Gamma_{0} \cup\Gamma_{1} \). The authors studied the uniform decay of solutions for a nonlinear viscoelastic wave equation with boundary dissipation. In [28], Boukhatem and Benabderrahmane have proved the existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions as follows:
where \(Lu =-\operatorname{div} (A\nabla u)=\sum_{i,j=1}^{N} \frac{\partial}{\partial x_{i} } ( a_{ij}(x) \frac{\partial u}{\partial x_{j} } )\) and \(\frac{\partial u}{\partial\nu_{L}}=\sum_{i,j=1}^{N} a_{ij}(x) \frac{\partial u}{\partial x_{j} } \nu_{i} \).
Liu [29] investigated the following viscoelastic wave equation with an interval time-varying delay term:
where Ω is a bounded domain \(R^{n}\) (\(n\geq2\)) with a boundary ∂Ω of class \(C^{2}\), α and g are positive non-increasing functions defined on \(R^{+}\), \(a_{0} \) and \(a_{1} \) are real number with \(a_{0} >0\), \(\tau(t)>0\) represents the time-varying delay. He also proved the general decay rate for the energy of a weak viscoelastic wave equation with an interval time-varying delay term.
Recently, Li and Chai [30] have investigated the energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback form:
where divX denotes the divergence of the vector field X in the Euclidean metric, \(A(x) =(a_{ij}(x))\) are symmetric and positive definite metrics for all \(x\in R^{n}\) and \(a_{ij}(x)\) are smooth functions on \(R^{n}\), \(\frac{\partial u}{\partial\nu_{A}}=\sum_{i,j=1}^{n} a_{ij} \frac{\partial u}{\partial x_{j}} \nu_{i}\), where \(\nu=(\nu_{1},\nu_{2}, \ldots, \nu_{n})^{T}\) denotes the outward unit normal vector of the boundary and \(\nu+A =A\mu\).
Motivated by previous work, in this paper, we study the energy decay rate of the nonlinear viscoelastic problem with a time-varying delay in the boundary conditions. Previously many authors have considered the uniform decay of solutions for a nonlinear viscoelastic wave equations with boundary dissipations. However, to our knowledge, there is no energy decay result of the nonlinear viscoelastic problem with a the dynamic, time-varying delay and acoustic boundary conditions. Thus this work is significant. The outline of the paper is the following. In Section 2, we give some notation and hypotheses for our result. In Section 3, we prove our main result.
2 Preliminary
In this section, we present some material that we shall use in order to present our results. We denote by \((u,v) =\int_{\Omega}u(x) v(x) \,dx \) the scalar product in \(L^{2} (\Omega)\). We denote by \(\|\cdot\|_{q} \) the \(L^{q} (\Omega)\) norm for \(1\leq q\leq\infty\) and \(\|\cdot\| _{q,\Gamma_{1}} \) for \(L^{q} (\Gamma_{1})\). We introduce by
the closed subspace of \(H^{1} (\Omega)\) equipped with the norm equivalent to the usual norm in \(H^{1} (\Omega)\). The Poincaré inequality holds in V, i.e., there exists a constant \(C_{*} \) such that
where
and there exists a constant \(\tilde{C_{*}}>0\) such that
For studying the problem (1.1)-(1.7) we will need the following assumptions:
- \((H1)\) :
-
The kernel function \(g: R^{+} \to R^{+} \) is a bounded \(C^{1} \) function satisfying
$$ g(0)>0,\quad \delta_{0}-\|a\|_{\infty} \int_{0}^{\infty}g(s) \,ds := l>0, $$(2.3)and there exists a non-increasing \(C^{1} \) positive differentiable function \(\zeta:R^{+} \to R^{+} \) satisfying
$$ g'(t) \leq-\zeta(t) g(t) ,\qquad \int_{0}^{\infty}\zeta(s) \,ds =\infty, \quad \forall t \geq0. $$(2.4) - \((H2)\) :
-
\(a : \Omega\to R \) is a nonnegative function and \(a\in C^{1} (\bar{\Omega}) \) such that
$$\begin{aligned}& a(x)\geq a_{0} >0, \end{aligned}$$(2.5)$$\begin{aligned}& \big\| \nabla a(x) \big\| ^{2} \leq\alpha_{1}^{2} \|a \|_{\infty}^{2} \end{aligned}$$(2.6)for some positive constant \(\alpha_{1} \), \(b: \Omega\to R \) is a nonnegative functions and \(b\in C^{1} (\bar{\Omega})\) such that \(b(x) \geq b_{0} >0 \).
- \((H3)\) :
-
Similarly to [31] \(k_{1} :R\to R \) is a nondecreasing \(C^{1} \) function such that there exist \(\varepsilon_{1}, C_{1} ,C_{2} >0 \) and a convex, increasing function \(K_{1} : R^{+} \to R^{+} \), of the class \(C^{1} (R^{+} )\cap C^{2} (R^{+})\) satisfying \(K_{1} (0)=0\), \(K_{1}\) is linear (or \((K_{1}'(0)=0)\) and \(K_{1} '' >0\) on \([0,\varepsilon_{1} ]\)) such that
$$\begin{aligned}& C_{1} |s| \leq\big|k_{1} (s)\big|\leq C_{2} |s| \quad\hbox{for all } |s|\geq \varepsilon_{1}, \end{aligned}$$(2.7)$$\begin{aligned}& s^{2}+ k_{1}^{2} (s) \leq K_{1}^{-1} \bigl(s k_{1} (s)\bigr) \quad \hbox{for all } |s|\leq \varepsilon_{1} , \end{aligned}$$(2.8)\(k_{2} : R \to R \) is an odd nondecreasing \(C^{1} \) function such that there exist \(C_{3}, C_{4}, C_{5} >0\),
$$\begin{aligned}& \big|k'_{2} (s)\big|\leq C_{3} , \end{aligned}$$(2.9)$$\begin{aligned}& C_{4} sk_{2} (s) \leq K_{2} (s) \leq C_{5} sk_{1} (s) \quad\hbox{for } s\in R, \end{aligned}$$(2.10)where
$$K_{2} (s) = \int_{0}^{s} k_{2} (r) \,dr. $$ - \((H4)\) :
-
For the time-varying delay, we assume as in [22] that \(\tau \in W^{2, \infty} ([0,T])\) for \(T>0\) and there exist positive constants \(\tau_{0}\), \(\tau_{1} \) such that
$$ 0< \tau_{0} \leq\tau(t) \leq\tau_{1},\quad \forall t>0. $$(2.11)Moreover, we assume that there exists \(d>0\) such that
$$ \tau' (t) \leq d < 1 \quad\hbox{for } t>0, $$(2.12)and that \(\mu_{1} \), \(\mu_{2} \) satisfy
$$ |\mu_{2} |< \frac{C_{4} (1-d)}{C_{5} (1-C_{4} d)}\mu_{1}. $$(2.13) - \((H5) \) :
-
The functions \(f, m, h >0\) are essentially bounded such that \(f(x), m(x), h(x) >0\). Furthermore, there exist positive constants \(f_{0} \), \(m_{0} \) and \(h_{0} \) such that
$$f(x) \geq f_{0} , \qquad m(x) \geq m_{0} ,\qquad h(x) \geq h_{0} \quad \hbox{for all a.e. } x\in\Gamma_{1} . $$
Remark 2.1
By the mean value theorem for integrals and the monotonicity of \(k_{2} \), we find that
Then from (2.10), we obtain \(C_{4} \leq1 \).
For studying problem (1.1)-(1.7), we introduce a new variable z as in [22],
Then problem (1.1)-(1.7) is equivalent to
Now we are in a position to state the local existence result of problem (2.15)-(2.23) which can be established by combining with the argument of [15, 32].
Theorem 2.1
Assume that \((H1)\)-\((H5)\) hold. Then given \((u_{0} , u_{1}) \in V \times L^{2} (\Omega)\), \(y_{0} \in L^{2} (\Gamma_{1}) \) and \(j_{0} \in L^{2} (\Gamma_{1} \times(0,1))\), there exist \(T>0\) and a unique weak solution \((u,y,z)\) of problem (2.15)-(2.23) such that
3 Global existence and asymptotic behavior
In order to study the global existence of solution for problem (2.15)-(2.23) given by Theorem 2.1, we define the functions
and
where \((g\circ\nabla u)(t)= \int_{\Omega}\int_{0}^{t} g(t-s) |\nabla u(t)-\nabla u(s)|^{2} \,ds \,dx \). Adopting the proof of [33], we still have the following results.
Lemma 3.1
For any \(u\in C^{1} (0,t;H^{1} (\Omega))\), we have
We define the modified energy functional \(E(t)\) associated with problem (2.15)-(2.23) by
where ξ is a positive constant such that
Lemma 3.2
Let \((u, y,z) \) be the solution of (2.15)-(2.23). Then the energy functional defined by (3.4) is a non-increasing function and for all \(t>0\), we have
Proof
Multiplying in (2.15) by \(u_{t} \), integrating over Ω, using Green’s formula and exploiting the conditions (2.16) and (2.17), we have
On the other hand, from (2.18), we see that
We also multiply the equation in (2.19) by \(\xi k_{2} (z(x,\rho ,t))\) and integrate over \(\Gamma_{1} \times(0,1) \) to obtain
and it follows that
Thus from (3.7), (3.8), (3.10) and using (2.10) and (3.3), we deduce
Let us denote by \(K_{2}^{*} \) the conjugate of the convex function \(K_{2} \), i.e.,
then
Moreover, \(K_{2} \) is the Legendre transform of \(K_{2} \) (thanks to the argument given in [34])
Then from the definition of \(K_{2} \) and (3.13), we get
Let us recall the following relations (Eq. (3.7) in [35]) derived from (3.14):
Using (3.11), (3.15) and (3.16), we obtain
Also using (2.10), (2.19), (3.16), this estimate becomes
Consequently, using (3.5), estimate (3.6) follows. Thus the proof of Lemma 3.2 is complete. □
Lemma 3.3
Let \((u, y, z)\) be the solution of (2.15)-(2.23). Assume that \(I(0)>0\) and
then \(I(t)>0 \) for all \(t\geq0\).
Proof
Since \(I(0)>0\), by continuity of \(u(t)\) there exists \(T_{*} < T\) such that \(I(t)>0\) for all \(t\in[0, T_{*} ]\). From (3.1), (3.2)
Hence from (2.3), (2.10) and the fact that \((g\circ \nabla u)(t)>0\), \(\forall t \geq0\), we can deduce
it follows that
Thus from (2.1), (3.17) and (3.19), we arrive at
Hence \(\|u(t)\|_{p}^{p} \leq C\|\nabla u(t)\|^{2} \), \(\forall t\in[0, T_{*})\), which implies that \(I(t)>0\), \(\forall\in[0, T_{*})\). Note that
We repeat the procedure with \(T_{*} \) extended to T. □
Theorem 3.1
Let \((u, y, z) \) be the solutions of problem (2.15)-(2.23). Suppose that (3.17) holds and \(I(0)>0\), then the solution \((u,y,z)\) is a global time.
Proof
It suffices to show that
is bounded independent of t. Under the hypotheses in Theorem 3.1, we see from Lemma 3.3 that \(I(t)>0\) on \([0,T]\). Using Lemma 3.2, from (3.18) it follows that
Thus, there exists a constant \(C>0\) depending p and l such that
Thus the proof of Theorem 3.1 is finished. □
4 General energy decay rate
In this section, we shall investigate the asymptotic behavior of the energy function \(E(t)\). For this purpose we construct a Lyapunov function \(\mathcal {L} (t)\) equivalent to \(E(t)\), which we can show to lead to the desired result. First, we define some functional and establish several lemmas. Let
where M and ε are positive constants to be chosen later and
and
The functional \(\mathcal {L} (t)\) is equivalent to the energy function \(E(t)\) by the following lemma.
Lemma 4.1
For \(\varepsilon>0\) small enough while M is large enough, there exist two positive constants \(\beta_{1} \) and \(\beta_{2} \) such that
Proof
From Hölder’s and Young’s inequality, (2.1), (2.2) and (4.2)-(4.4) we have
where we used
and
where C is a positive constant. Combining (4.1) and (4.6)-(4.8), then we arrive at
where C is a positive constant. Choosing \(M>0\) large, we complete the proof of Lemma 4.1. □
Lemma 4.2
Let \((u,y,z)\) be the solution of (2.15)-(2.23). Then the functional Ψ defined in (4.2) satisfies
Proof
Taking the derivatives of \(\Psi(t)\) defined in (4.2) and using (2.15)-(2.18) we have
Now, by using Hölder’s and Young’s inequality, \((H1)\), (2.1) and (2.2) we estimate the right hand side of (2.10) as follows, for any \(\eta>0\):
and
Thus from (4.10)-(4.16) we conclude that
Thus we finished the proof of Lemma 4.2. □
Lemma 4.3
Let \((u,y,z)\) be the solution of problem (2.15)-(2.23). Then the functional \(\Phi(t)\) defined in (4.3) satisfies
Proof
Taking the derivative of \(\Phi(t)\) defined in (4.3) and using (2.15)-(2.18), we have
Similarly to (4.9), we estimate each terms in the right hand side of (4.19). Using Hölder’s and Young’s inequality, \((H1)\), \((H2)\), (2.1), (2.2), (2.3), (2.5), (2.6) and (3.19), for any \(\eta>0\), we have
Combining the estimates (4.20)-(4.30), then (4.19) becomes
□
Lemma 4.4
Let \((u,y,z)\) be the solution of problem (2.15)-(2.23). Then the functional \(\Lambda(t)\) defined in (4.4) satisfies
Proof
Multiplying (2.19) by \(e^{-\rho\tau(t)}k_{2} (z(x,\rho, t))\) and integrating over \(\Gamma_{1} \times(0,1)\), we obtain
Differentiating (4.4) with respect to t and using (4.32), we get
Then, by integration by parts and using (2.12), (4.33) lead to
Using (2.10), then (4.31) holds. □
Now we are in a position to state our main result.
Theorem 4.1
Assume that (H1)-(H5) and (3.5) hold. Then, for each \(t_{0} >0\), there exist positive constants θ, \(\theta_{1} \), \(\theta_{2}\) and \(\varepsilon_{0} \) such that the solution energy of (2.15)-(2.23) satisfies
where
and
Proof
Since the function \(g(t)\) is positive, there exists \(t_{0} >0\) such that
Using (3.6), (4.9), (4.17) and (4.31), we arrive at
At this point, we choose \(\varepsilon>0 \) small enough and we pick \(\eta>0 \) sufficiently small and M is so large such that
Therefore, for all \(t\geq t_{0} \), we deduce
where
which implies that
where \(M_{10}\) and \(M_{11}\) are some positive constants.
Multiplying the above inequality by \(\zeta(t) \), we obtain, for any \(t\geq t_{0}\),
From (2.9), we obtain
where c is some positive constant. Recalling (2.4) and (3.6), we get for any \(t\geq t_{0} \)
Now, we define
As ζ is a non-increasing positive function, by using Lemma 4.1, the function \(G(t)\) is equivalent to \(E(t)\). Using the fact that \(\zeta'(t)\leq0\), (4.34) implies that
In the following, we shall estimate the term \(\int_{\Gamma_{1}} k_{1}^{2} (u_{t} (t)) \,d\Gamma\) in (4.35). To do this, let \(\Gamma_{11} =\{ x\in\Gamma_{1} : |u_{t} |>\varepsilon _{1}\}\), \(\Gamma_{12} =\{ x\in\Gamma_{1} : |u_{t} |\leq\varepsilon_{1} \} \).
Case (I): \(K_{1}\) is linear on \([0, \varepsilon_{1}] \).
There exist positive constants \(C_{1} \) and \(C_{2}\) such that
This and (4.35) yield
where \(M_{13}\) is a positive constant. This gives
Employing that G is equivalent to E, we get
where C and ν are positive constants. Owing to \(K_{1} (s) = \sqrt {s} k_{1} (\sqrt{s}) =cs \),
Case (II): \(K_{1}'(0) =0 \) and \(K_{1} '' (t) >0 \) on \([0, \varepsilon _{1}] \).
Since \(K_{1} \) is convex and increasing, \(K_{1}^{-1}\) is concave and increasing. By (2.7), (2.8), (3.6) and the reversed Jensen inequality
where \(\mu_{3} \) and \(\mu_{4}\) are positive constants. Thus, we get from (4.35)
where \(G_{1}(t) = G(t) + \mu_{3} M_{12} E(t) \). Now, for \(0 < \varepsilon _{0} < \varepsilon_{1}\) and \(\mu>0\), we define
It is easily noted that
where \(\gamma_{1}\), \(\gamma_{2} \) are positive constant. Thanks to the similar argument in (3.13) and (3.12), we have
where \(M_{13} =|\Gamma_{12} | M_{12} \mu_{4} \zeta(0) \) and \(M_{14} = |\Gamma_{12}| M_{12} \). Taking \(\varepsilon_{0} \) sufficiently small such that \(M_{10} E(0) - \varepsilon_{0} M_{14} >0 \) and \(\mu>0 \) suitably so that \(\mu- M_{13} >0 \), we arrive at
where \(O_{1} (t)=tK_{1}' (\varepsilon_{0} t)\) and \(\mu_{5} \) is a positive constant. Finally, we define
Using (4.37), we see that \({\mathcal {E} } \) is equivalent to E. Therefore,
for some \(\theta_{1} >0 \), and
Thus the proof is completed. □
5 Conclusion
We have investigated the energy decay rate of the nonlinear viscoelastic problem with dynamic and acoustic boundary conditions.
It is well known that viscoelastic materials have memory effects, which is due to the mechanical response influenced by the history of the materials themselves. As these materials have a wide application in the natural sciences, their dynamics is interesting and of great importance. Also, the dynamic boundary conditions are not only important from the theoretical point of view but also arise in numerous practical problems and the acoustic boundary conditions are related to noise control and suppression in practical applications. Moreover, time delay so often arises in many physical, chemical, biological and economical phenomena because these phenomena depend not only on the present state but also on the history of the system in a more complicated way. We established a decay rate estimate for the energy by introducing suitable Lyapunov functionals.
References
Cavalcanti, MM, Domingos Cavalcanti, VN, Soriano, JA: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differ. Equ. 2002, Article ID 44 (2002)
Berrimi, S, Messaoudi, SA: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differ. Equ. 2004, Article ID 88 (2004)
Cavalcanti, MM, Oquendo, HP: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310-1324 (2003)
Cavalcanti, MM, Domingos Cavalcanti, VN, Prates Filho, JS, Soriano, JA: Existence and uniform decay rate for viscoelstic problems with nonlinear boundary damping. Differ. Integral Equ. 14(1), 85-116 (2001)
Liu, W, Sun, Y, Li, G: On decay and blow-up of solutions for a singular nonlocal viscoelastic problem with a nonlinear source term. Topol. Methods Nonlinear Anal. 49(1), 299-323 (2017)
Hao, J, Cai, L: Uniform decay of solutions for coupled viscoelastic wave equations. Electron. J. Differ. Equ. 2016, Article ID 72 (2016)
Messaoudi, S, Mukiawa, SE: Existence and decay of solutions to a viscoelastic plate equation. Electron. J. Differ. Equ. 22, 14 (2016)
Grobbelaar-Van Dalsen, M: On fractional powers of a closed pair of operators and a damped wave equation with dynamic boundary conditions. Appl. Anal. 53(1-2), 41-54 (1994)
Grobbelaar-Van Dalsen, M: On the initial-boundary-value problem for the extensible beam with attached load. Math. Methods Appl. Sci. 19(12), 943-957 (1996)
Cavalcanti, MM, Khemmoudj, A, Medjden, M: Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions. J. Math. Anal. Appl. 328, 900-930 (2007)
Ferhat, M, Hakem, A: Global existence and energy decay result for a weak viscoelastic wave equations with a dynamic boundary and nonlinear delay term. Comput. Math. Appl. 71, 779-804 (2016)
Gerbi, S, Said-Houari, B: Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions. Nonlinear Anal. 74, 7137-7150 (2011)
Liu, W, Zhu, B, Li, G, Wang, D: General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and time-varying delay term. Evol. Equ. Control Theory 6(2), 239-260 (2017)
Morse, PM, Ingard, KU: Theoretical Acoustics. McGraw-Hill, New York (1968)
Beale, JT, Rosencrans, SI: Acoustic boundary conditions. Bull. Am. Math. Soc. 80, 1276-1278 (1974)
Liu, W, Sun, Y: General decay of solutions for a weak viscoelastic equation with acoustic boundary conditions. Z. Angew. Math. Phys. 65, 125-134 (2014)
Liu, W: Arbitrary rate of decay for a viscoelastic equation with acoustic boundary conditions. Appl. Math. Lett. 38, 155-161 (2014)
Lee, MJ, Park, JY, Kang, YH: Exponential decay rate for a quasilinear von Karman equation of memory type with acoustic boundary conditions. Bound. Value Probl. 2015, Article ID 122 (2015)
Lee, MJ, Kim, DW, Park, JY: General decay of solutions for Kirchhoff type containing Balakrishnan-Taylor damping with a delay and acoustic boundary conditions. Bound. Value Probl. 2016, Article ID 173 (2016)
Mustafa, MI: Asymptotic behavior of second sound thermoelasticity with internal time-varying delay. Z. Angew. Math. Phys. 64, 1353-1362 (2013)
Lee, MJ, Park, JY, Kang, YH: Asymptotic stability of a problem with Balakrishnan-Taylor damping and a time delay. Comput. Math. Appl. 70, 478-487 (2015)
Nicaise, S, Pignotti, C: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45, 1561-1585 (2006)
Nicaise, S, Pignotti, C: Stability of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 2011, Article ID 41 (2011)
Nicaise, S, Pignotti, C: Interior feedback stabilization of wave equations with time dependent delay. Electron. J. Differ. Equ. 2011, Article ID 41 (2011)
Grace, S: Oscillation criteria for third order nonlinear delay differential equations with damping. Opusc. Math. 35(4), 485-497 (2015)
Malygina, V, Sabatulina, T: On oscillation of solutions of differential equations with distributed delay. Electron. J. Qual. Theory Differ. Equ. 116, 15 (2016)
Wu, ST, Chen, HF: Uniform decay of solutions for a nonlinear viscoelastic wave equation with boundary dissipation. J. Funct. Spaces Appl. 2012, Article ID 421847 (2012)
Boukhatem, Y, Benabderrahmane, B: Existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions. Nonlinear Anal. 97, 191-209 (2014)
Liu, W: General decay rate estimate for the energy of a weak viscoelastic equation with an internal time-varying delay term. Taiwan. J. Math. 17(6), 2101-2115 (2013)
Li, J, Chai, S: Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback. Nonlinear Anal. 112, 105-117 (2015)
Lasiecka, I, Tataru, D: Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6, 507-533 (1993)
Benaissa, A, Benaissa, A, Messaoudi, SA: Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks. J. Math. Phys. 53, Article ID 123514 (2012)
Li, F, Zhao, Z, Chen, Y: Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation. Nonlinear Anal. 12(3), 1759-1773 (2011)
Arnold, VI: Mathematical Methods of Classical Mechanics. Springer, New York (1989)
Park, SH: Energy decay for a von Karman equation with time-varying delay. Appl. Math. Lett. 55, 10-17 (2016)
Acknowledgements
The authors wold like to express the their gratitude to the anonymous referees for a helpful and very careful reading of this paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2017R1E1A1A03070738).
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Lee, M.J., Park, J.Y. Energy decay of solutions of nonlinear viscoelastic problem with the dynamic and acoustic boundary conditions. Bound Value Probl 2018, 1 (2018). https://doi.org/10.1186/s13661-017-0918-2
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DOI: https://doi.org/10.1186/s13661-017-0918-2