Abstract
In this paper we consider the existence and general energy decay rate of global solution to the mixed problem for the Kirchhoff-type equation with memory boundary and acoustic boundary conditions. In order to prove the existence of solutions, we employ the Galerkin method and compactness arguments. Besides, we establish an explicit and general decay rate result using the perturbed modified energy method and some properties of the convex functions. Our result is obtained without imposing any restrictive assumptions on the behavior of the relaxation function at infinity. These general decay estimates extend and improve some earlier results, i.e., exponential or polynomial decay rates.
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1 Introduction
In this paper, we are concerned with the general decay of solutions to the Kirchhoff-type equation with memory boundary and acoustic boundary conditions:
where Ω is a bounded domain of \({\mathbb {R}}^{n}\) with sufficiently smooth boundary Γ, ν represents the outward unit normal vector to \(\Gamma= \Gamma_{0} \cup\Gamma_{1}\) for \(\Gamma_{0}\) and \(\Gamma_{1}\) be closed and disjoint. The relaxation function g is positive and nondecreasing, the function \(f \in C^{1} ({\mathbb {R}})\) and \(M \in C^{1} ([0, \infty[)\), and p, q are functions satisfying some conditions to be specified later.
On the other hand, problem (1.1) with \(u=0\) on ∂Ω has its origin in the mathematical description of small amplitude vibrations of an elastic string. The existence of global solutions and exponential decay to this problem has been studied by many authors (see [1–4]). In fact, a mathematical model for the deflection of an elastic string of length \(L>0\) is given by the mixed problem for the nonlinear wave equation
where u is the lateral deflection, x is the space coordinate, t is the time, ρ denotes the mass density, h is the cross section area, \(p_{0}\) is the initial axial tension and E is the Young modulus. Eq. (1.6) was introduced by Kirchhoff [5] as a nonlinear model of the free transversal vibrations of a clamped string.
The asymptotic behavior of solutions for nonlinear wave and plate equations with memory boundary condition has been proved by many authors [6–12]. In the aforementioned results, denoting by k the resolvent kernel of \({- {g' \over g(0)},}\) they showed that the energy of the solution decays exponentially (polynomially) to zero provided k decays exponentially (polynomially) to zero. The decay result of Santos [9] was generalized by Messaoudi and Soufyane [13] without assuming the exponential (polynomial) decay of k. They obtained general stability for a wave equation under weaker condition on the resolvent kernel k such as
where γ is a nonincreasing and positive function. Kang [14], Mustafa and Messaoudi [15] and Santos and Soufyane [16] investigated the general decay for the Kirchhoff plates, the Timoshenko system and the von Karman plate system with viscoelastic boundary conditions under condition (1.7), respectively. Recently, Kang [17] established a more general decay result of the differential inclusion of Kirchhoff type with strong damping term and boundary condition of memory type when a relaxation function satisfies the condition (1.7). This result improved the earlier decay results of Santos et al. [11]. More precisely, we studied that the energy decays at the rate similar to the relaxation functions, which are not necessarily decaying like polynomial or exponential functions.
Moreover, Beale and Rosencrans [18] introduced acoustic boundary conditions of the general form, and then Beale [19, 20] proved global existence and regularity of solutions for wave equations with acoustic boundary conditions. In these cases, the solution u of the wave equation is the velocity potential of a fluid undergoing acoustic wave motion and y is the normal displacement to the boundary at time t with the boundary point x. Recently, wave equations with acoustic boundary conditions have been treated by many authors [21–24]. They considered the existence of solutions, but gave no decay rate for solutions. As regards uniform decay rates for solutions to problems with acoustic boundary conditions, there is not much literature [25–29]. Most of these are concerned with exponential decay rates of solutions.
Motivated by these results, we study the stability for the Kirchhoff-type equation (1.1), which contains both memory boundary conditions and acoustic boundary conditions for resolvent kernel k satisfying
where H is a positive function, with \(H(0)=H'(0) =0\), and H is linear or strictly increasing and strictly convex on \((0, r]\) for some \(0< r<1\). Recently, Mustafa and Abusharkh [30] and Kang [31] showed the general decay result for plate equations and von Karman plate system with viscoelastic boundary damping when a relaxation function satisfies (1.8) and \(u_{0} \equiv0\) on \(\Gamma_{0}\), respectively. We obtain an explicit and general decay of the solution for the Kirchhoff-type equation without assuming that \(u_{0} \equiv0\) on \(\Gamma_{0}\) when relaxation function satisfies (1.8). Since problem (1.1) does not have a homogeneous Dirichlet condition on portion of the boundary, we introduce a close subspace Ṽ of \(H^{1}(\Omega)\), as in [24], where Poincaré’s inequality is satisfied. Moreover, to prove the existence of a weak solution to the problem, we use the Galerkin method and compactness arguments. After this, we obtain the general decay rates by employing the multiplier method and some properties of convex functions including the use of general Young’s inequality and Jensen’s inequality.
The paper is organized as follows. In Section 2 we give some notations and material needed for our work and state the main results. In Section 3 we consider the existence of global weak solution for problem (1.1)-(1.5). In Section 4 we show the general decay of the solutions to the Kirchhoff-type equation with memory boundary and acoustic boundary conditions.
2 Statement of main results
In this section, we provide some material needed in the proof of our main result and state main results. Let us consider the Hilbert spaces \(L^{2}(\Omega)\) and \(L^{2} (\Gamma)\) endowed with the inner products
and the corresponding norms \(\Vert u\Vert ^{2}_{L^{2}(\Omega)}=(u, u)\) and \(\Vert u\Vert ^{2}_{L^{2}(\Gamma)}=(u, u)_{\Gamma}\), respectively. For simplicity, we denote \(\Vert \cdot \Vert ^{2}_{L^{2}(\Omega)}\) and \(\Vert \cdot \Vert ^{2}_{L^{2}(\Gamma)}\) by \(\Vert \cdot \Vert \) and \(\Vert \cdot \Vert _{\Gamma}\), respectively.
Following the idea in [24], we consider
where, for each point \(x_{0}\) fixed in Γ,
Then Poincaré’s inequality holds in V. From density, we see that Poincaré’s inequality still holds in \(H^{1}(\Omega)\) closure of V which we denote by \(W=\bar{V}^{H^{1}(\Omega)}\). Let λ and \(\lambda_{1}\) be the smallest positive constants such that
Moreover, we let \(x^{0}\) be a fixed point in \({\mathbb {R}}^{n}\), \(m(x) = x - x^{0} \) and \(R = \max\{ \vert x - x^{0} \vert : x \in\bar{\Omega} \}\), and assume that
We formulate the following hypotheses.
-
(H1)
With respect to \(M \in C^{1} ([0, \infty[ )\), we assume that
$$ 0< m_{0} \leq M(\zeta) ,\quad \quad \hat{M}(\zeta) \leq M(\zeta) \zeta ,\quad \forall \zeta\geq0, $$(2.3)where \(\hat{M}(\zeta)= {\int_{0}^{\zeta}M(s) \,ds}\).
-
(H2)
Let \(f \in C^{1} ( {\mathbb {R}})\) satisfy \(f(s) s \geq0\), \(\forall s \in {\mathbb {R}}\). We suppose that f is superlinear, that is,
$$ f(s) s \geq(2+ \delta) F(s), \quad\quad F(z) = \int_{0}^{z} f(s) \,ds, \quad \forall s \in{\mathbb {R}} $$(2.4)for some \(\delta>0\) with the following growth condition:
$$ \bigl\vert f(x) - f(y) \bigr\vert \leq c_{0} \bigl( 1 + \vert x \vert ^{\rho-1} + \vert y\vert ^{\rho-1} \bigr) \vert x-y \vert , \quad \forall x, y \in{\mathbb {R}} $$(2.5)for some \(c_{0}>0\) and \(\rho \geq1\) such that \((n-2) \rho\leq n \).
-
(H3)
For the functions p and q, we assume that \(p, q \in C(\Gamma_{1})\) and \(p(x) >0\) and \(q(x)>0\) for all \(x \in \Gamma_{1}\). It implies that there exist positive constants \(p_{i}\), \(q_{i}\) (\(i=0, 1\)) such that
$$ p_{0} \leq p(x) \leq p_{1}, \quad\quad q_{0} \leq q(x) \leq q_{1} , \quad \forall x \in\Gamma_{1}. $$(2.6) -
(H4)
In addition, we assume that \(k : {\mathbb {R}}_{+} \rightarrow{\mathbb {R}}_{+}\) is the resolvent kernel of \(-\frac{g'}{g(0)}\), which is a twice differentiable function such that
$$ k(0)>0 , \quad\quad \lim_{t\rightarrow\infty} k(t) =0, \quad\quad k' (t)\leq 0, $$(2.7)and there exists a positive function \(H \in C^{1} ({\mathbb {R}}_{+})\) and H is a linear or strictly increasing and strictly convex \(C^{2}\) function on \((0, r]\), \(r<1\), with \(H(0)=H'(0)=0\), such that
$$ k''(t) \geq H \bigl(- k' (t) \bigr) ,\quad t > 0. $$(2.8)
To simplify calculation in our analysis, we introduce the following notation:
First, we shall use Eq. (1.2) to estimate the term \(M(\Vert \nabla u\Vert ^{2} ) {\frac{\partial u}{\partial\nu}+ \frac{\partial u'}{\partial\nu}}\). Differentiating Eq. (1.2), we get the following Volterra equation:
Using the Volterra inverse operator, we have
where the resolvent kernel k is given by the solution of
Denoting \(\tau= {{1 \over {g(0)}}}\), we obtain
which is equivalent to condition (1.2). Then we get the following equivalent problem:
By differentiating the term \(g \square v\), we have the following lemma.
Lemma 2.1
If \(g, v \in C^{1} ( [ 0, \infty): {\mathbb {R}} )\), then
The energy of system (2.9)-(2.13) is given by
Now, we are ready to state our main results.
Theorem 2.1
Suppose that (H1)-(H4) hold. If \(( u_{0} , u_{1} ) \in(W \cap H^{2}(\Omega)) \times W\) and satisfy the compatibility condition
then, for all \(T>0\), there exists a unique pair of functions \((u, y)\), which is a solution of system (2.9)-(2.13) satisfying
Theorem 2.2
Suppose that (H1)-(H4) hold. Assume that D is a positive \(C^{1}\) function, with \(D(0)=0\), for which \(H_{0}\) is a strictly increasing and strictly convex \(C^{2}\) function on \((0, r]\) and
Therefore, there exist positive constants \(c_{1}\), \(c_{2}\), \(c_{3}\) and \(\epsilon_{0}\) such that the solution of (2.9)-(2.13) satisfies
where
Furthermore, if \({\int_{0}^{1} H_{1} (t) \,dt } < +\infty\), for some choice of D, then we obtain
where
In particular, (2.17) is valid for the special case \(H(t) = c t^{p}\) for \(1\leq p< \frac{3}{2}\).
Remark 2.1
For large \(t_{0}>0\), there exists a constant \(d_{0} >0\) such that
Indeed, from (H4), we find that \({\lim_{t \rightarrow +\infty} (-k' (t)) =0}\). This implies that \({\lim_{t \rightarrow+\infty} k'' (t)}\) cannot be equal to a positive number, and so it is natural to assume that \({\lim_{t \rightarrow+\infty} k'' (t) =0}\). Then there is \(t_{0} >0\) large enough such that \(k'(t_{0}) <0\) and
Because \(k'\) is nondecreasing, \(k'(0)<0\) and \(k'(t_{0} ) <0\), we get
From H is a positive continuous function, we have for some positive constants \(d_{1}\) and \(d_{2}\),
Therefore, by (2.8), (2.21) and (2.22), we see that (2.19) holds.
The well-known Jensen’s inequality will be of essential use in establishing our main result.
Remark 2.2
If \(F_{0}\) is a convex function on \([a, b], f:\Omega\rightarrow[a, b]\) and h are integrable functions on Ω, \(h(x)\geq0\), and \(\int_{\Omega}h (x) \,dx = h_{0} >0\), then Jensen’s inequality states that
3 Proof of Theorem 2.1
In this section, we study the existence of a global weak solution for problem (2.9)-(2.13) using Faedo-Galerkin’s approximation. Since the problem is not normal, we cannot apply directly Caratheodory’s theorem. So we use a degenerated second order equation on \(\Gamma_{1}\). To this end, let \(\{w_{j}\}_{j\in{\mathbb {N}}}\) and \(\{z_{j} \}_{j\in{\mathbb {N}}}\) be orthonormal bases of W and \(L^{2}(\Gamma)\), respectively. For each \(m\in{\mathbb {N}}\), let \(W_{m} =\operatorname{span}\{w_{1}, w_{2}, \ldots, w_{m} \} \) and \(Z_{m} =\operatorname{span}\{z_{1}, z_{2}, \ldots, z_{m} \} \). For each \(\epsilon\in(0, 1)\) and any \(T<0\), standard results on ordinary differential equations guarantee that there exists only one local solution for \(0< T_{m} \leq T\)
satisfying the approximate perturbed problem
for \(j=1, 2, \ldots, m\). Now we need estimates which allow us to extend the solutions to the whole interval \([0, T]\) and pass to limit as \(m\rightarrow\infty\) and \(\epsilon\rightarrow0\). Hence, uniform estimates with respect to m and ϵ are needed. Indeed, from (3.1), we obtain the approximate equations
Estimate I
Taking \(w=u_{m\epsilon}'\) and \(z=y_{m\epsilon}'\) in (3.2) and integrating over \((0, t)\), we get from Lemma 2.1
Using Young’s inequality and (2.7), we have
From (2.6)-(2.8), (3.3) and (3.4), we obtain
Then, employing Gronwall’s inequality, we conclude that there exists a constant \(C=C(T)\), independent of m, ϵ and \(t\in[0, T]\), such that
Estimate II
First, we will estimate \(\Vert u_{m \epsilon}''(0)\Vert ^{2} \) and \(\Vert y_{m \epsilon}''(0)\Vert ^{2}_{\Gamma_{1}}\). Taking \(t=0\) in (3.2), replacing w and z by \(u_{m \epsilon}'' (0)\) and \(y_{m \epsilon}''(0)\), respectively, and using (2.14), we get
and
From the assumptions on the initial data, f and M, we have that there exists a constant \(C>0\), independent of ϵ and m, such that
Differentiating (3.2) with respect to t and substituting w and z by \(u_{m \epsilon}''\) and \(y_{m \epsilon}''\), respectively, we see that
From the first estimate, assumption on M and Young’s inequality, we obtain
and
Using generalized Hölder’s inequality, assumption (2.5), (3.5), the Sobolev imbedding and Young’s inequality, we find that
Noting that
and using Lemma 2.1, we get
Combining (3.10)-(3.13) with (3.9), we deduce that
Integrating (3.14) over \([0, t]\) and applying Gronwall’s inequality and (H4), we conclude that there exists a constant C, independent of ϵ and m, such that
From (3.5) and (3.15) and Lions-Aubin’s compactness theorem [32], we can pass to the limit in (3.1). This completes the proof of Theorem 2.1.
4 Proof of Theorem 2.2
In this section, we shall prove the general decay rates in Theorem 2.2. Let us consider the following binary operator:
Then, using Hölder’s inequality for \(0 \leq\alpha\leq1\), we have
Lemma 4.1
The energy E satisfies, along the solution of (2.9)-(2.13),
Proof
Multiplying Eq. (2.9) by \(u'\) and integrating by parts over Ω, we obtain
From Lemma 2.1 and Young’s inequality, we get estimate (4.2). □
To this system, we introduce the functional
where θ is a small positive constant. The following lemma plays an important role in the construction of the Lyapunov functional.
Lemma 4.2
There exists \(C>0\) such that
Proof
Direct computations and (2.9) yield
Integrating by parts and using Young’s inequality, we have
We know that
From (4.5), the boundary condition (2.10) can be written as
Applying Young’s and Poincaré’s inequalities, (2.11), (4.6) and (4.1) with \(\alpha=\frac{1}{2}\), we obtain, for \(\epsilon_{1}>0\),
By (2.1), (2.3) and Young’s inequality, we get, for \(\epsilon_{2}>0\),
and
Substituting (4.7)-(4.9) into (4.4) and using (2.4), we deduce that
Using (2.2) and choosing \(\epsilon_{1}\) small enough, we have estimate (4.3). □
Proof of Theorem 2.2
Let us introduce the Lyapunov functional
with \(N >0\). From (4.2) and (4.3), we obtain, for all \(t\geq t_{0}\),
We take θ, ϵ and \(\epsilon_{2} >0\) so small that
And then, choosing N large, for some positive constant \(\theta_{0}\), we have
which, using the fact that \(\lim_{t\rightarrow\infty} k(t) =0\), yields, for large \(t_{0}\),
Meanwhile, we can choose N even larger so that
Therefore, from (2.19), (4.2) and (4.10), we get
We take \({\mathcal {L}}(t) = L(t) +\frac{2C}{d_{0} \tau}E(t)\), which is clearly equivalent to \(E(t)\). Then by (4.12) we arrive at
(A) The special case \(H(t) =ct^{p} \) and \(1\leq p< \frac{3}{2}\):
Case 1. \(p=1\): From (2.8), (4.2) and (4.13), we have
which yields
By (4.11), we find that \({\mathcal {L}}+ \frac{2C}{c\tau} { E} \sim E\). Then we easily obtain
where
Case 2. \(1< p<\frac{3}{2}\): We see that
for any \(\delta_{0} <2-p\). Using (4.14) and taking \(t_{0}\) even larger if needed, we get, for all \(t \geq t_{0}\),
From Hölder’s inequality, (2.8), (2.23), (4.2) and (4.15), we deduce that
Hence, by (4.16), estimate (4.13) yields, for \(\delta_{0} =\frac{1}{2}\),
Multiplying (4.17) by \({ E}^{\gamma}(t)\), with \(\gamma=2p-2\), and using Young’s inequality, we have
where we have used the fact \(E'(t)\leq0\), \(\forall t\geq t_{0}\). Then, taking \(\varepsilon<\theta_{0} \), we obtain, for some \(C_{1}>0\),
where \(L_{0} ={\mathcal {L} }{ E}^{\gamma}+C_{\varepsilon}{ E} \sim E \). Hence we get
On the other hand, using (4.18), we have, for \(p<\frac{3}{2}\),
Therefore, by Hölder’s inequality, (2.8), (4.2) and (4.19), estimate (4.13) yields
Then, multiplying (4.20) by \({ E}^{\gamma}(t)\) with \(\gamma=p-1\) and repeating the above steps, we see that
where
(B) The general case: Because of the ideas presented in [30, 31, 33], this case is obtained as follows. We define \(\eta(t)\) by
where \(H_{0}\) satisfies (2.15). Like in (4.15), we see that \(\eta(t)\) satisfies
Moreover, we define \(\kappa(t)\) by
Because \(H_{0} (0)=0\) and \(H_{0}\) is strictly convex on \((0, r]\), then
provided \(0\leq \lambda\leq1\) and \(x\in(0, r]\). From (2.23), (4.21) and (4.22), we get
Indeed,
Then by (4.23) estimate (4.13) yields
Now, for \(\epsilon_{0} < r\) and \(\alpha_{0} > 0\), we define the functional
which satisfies, for some \(\alpha_{1}, \alpha_{2}>0\),
By using a similar analysis as in [30, 31], we can compute to find
Therefore, with a suitable choice of \(\epsilon_{0}\) and \(\alpha_{0}\), we have, for all \(t \geq t_{0}\),
where \(\alpha_{3}>0\) and \(H_{2} (t) = t H_{0}' (\epsilon_{0} t)\). From
and the strict convexity of \(H_{0}\) on \((0, r]\), we find that \(H_{2} ' (t) , H_{2} (t) >0\) on \((0, 1]\). We denote
which is clearly equivalent to \({E}(t)\). From (4.25) and (4.26), we obtain
where \(k_{0} = \frac{ \alpha_{1} \alpha_{3}}{{ E}(0)} >0 \). Consequently, a simple integration gives, for some \(k_{1}, k_{2}>0\),
where \(H_{1} (t) =\int_{t}^{1} \frac{1}{H_{2} (s)} \,ds\). Here, we have used the properties of \(H_{2}\) and the fact that \(\lim_{t\rightarrow0} H_{1} (t)= +\infty\) and \(H_{1}\) is a strictly decreasing function on \((0, 1]\). Using (4.27), we see that (2.16) holds.
Furthermore, if \(\int_{0}^{t} H_{1} (t) \,dt < +\infty\), then \(\int_{0}^{+\infty} H_{1} ^{-1} (t)\,dt < +\infty\). From (2.16), we get \(\int_{0}^{+\infty} E(t) \,dt < \infty\) and
Analogously, we define, for large \(t_{0}\),
and
Using (2.8), (2.23) and the strict convexity of H, we find that
Indeed,
Hence, by (4.28), estimate (4.13) becomes
Consequently, repeating the same procedures, we deduce that for some \(c_{1}\), \(c_{2}\) and \(c_{3} >0\),
where \(G(t) = \int_{t}^{1} \frac{1}{sH'(\epsilon_{0} s)} \,ds \). □
Examples
We give some examples to explain the energy decay rates given by Theorem 2.2.
(1) As in [30], let \(0< q<1\)
then \(k''(t)=H(-k'(t))\), where \(H(t)=\frac{qt}{[\ln(1/t)]^{ \frac{1}{q}-1}}\) for \(t\in(0,r]\), \(r<1\). Therefore,
(2) As in [31], if
for \(q>3\) and \(a>1\), then \(k''(t) =H(-k'(t))\), where
Since
then the function H satisfies hypothesis (H4) on the interval \((0, r]\) for any \(0< r<\frac{1+q-\sqrt{q^{2}-1}}{2aq}\). By taking \(D(t)=t^{\alpha}\), (2.15) is satisfied for any \(\alpha>\frac{q}{q-1}\). Then an explicit rate of decay can be obtained by Theorem 2.2. The function \(H_{0} (t) = H(t^{\alpha})\) has derivative
Therefore,
Now, we see that if \(\alpha< \frac{2q}{1+q}\),
Choosing \(\frac{1}{\epsilon_{0} s} =v \) and \(\epsilon_{0} < a^{-1}\), we have
Hence,
Consequently, we can use (2.17) to conclude that the energy decays
where \(\tilde{c}_{i}\) (\(i=1, 2 , 3\)) are constants.
References
Ikehata, R: On solutions to some quasilinear hyperbolic equations with nonlinear inhomogeneous terms. Nonlinear Anal. 17(2), 181-203 (1991)
Ikehata, R: A note on the global solvability of solutions to some nonlinear wave equations with dissipative terms. Differ. Integral Equ. 8(3), 607-616 (1995)
Park, JY, Bae, JJ: On the existence of solutions of the degenerate wave equations with nonlinear damping terms. J. Korean Math. Soc. 35(2), 465-489 (1998)
Park, JY, Bae, JJ, Jung, IH: Uniform decay of solution for wave equation of Kirchhoff type with nonlinear boundary damping and memory term. Nonlinear Anal. 50, 871-884 (2002)
Kichhoff, G: Vorlesungen über Mechanik. Teubner, Leipzig (1883)
Cavalcanti, MM, Domingos Cavalcanti, VN, Santos, ML: Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary. Appl. Math. Comput. 150(2), 439-465 (2004)
Rivera, JEM, Andrade, D: Exponential decay of non-linear wave equation with a viscoelastic boundary condition. Math. Methods Appl. Sci. 23, 41-61 (2000)
Rivera, JEM, Oquendo, HP, Santos, ML: Asymptotic behavior to a von Kármán plate with boundary memory conditions. Nonlinear Anal. 62, 1183-1205 (2005)
Santos, ML: Asymptotic behavior of solutions to wave equations with a memory condition at the boundary. Electron. J. Differ. Equ. 2001, 73 (2001)
Santos, ML: Decay rates for solutions of a Timoshenko system with a memory condition at the boundary. Abstr. Appl. Anal. 7(10), 531-546 (2002)
Santos, ML, Ferreira, J, Pereira, DC, Raposo, CA: Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary. Nonlinear Anal. 54, 959-976 (2003)
Santos, ML, Junior, F: A boundary condition with memory for Kirchhoff plates equations. Appl. Math. Comput. 148, 475-496 (2004)
Messaoudi, SA, Soufyane, A: General decay of solutions of a wave equation with a boundary control of memory type. Nonlinear Anal., Real World Appl. 11, 2896-2904 (2010)
Kang, JR: General decay for Kirchhoff plates with a boundary condition of memory type. Bound. Value Probl. 2012, 129 (2012)
Mustafa, MI, Messaoudi, SA: Energy decay rates for a Timoshenko system with viscoelastic boundary conditions. Appl. Math. Comput. 218, 9125-9131 (2012)
Santos, ML, Soufyane, A: General decay to a von Karman plate system with memory boundary conditions. Differ. Integral Equ. 24(1-2), 69-81 (2011)
Kang, JR: General decay for a differential inclusion of Kirchhoff type with a memory condition at the boundary. Acta Math. Sci. 34B(3), 729-738 (2014)
Beale, JT, Rosencrans, SI: Acoustic boundary conditions. Bull. Am. Math. Soc. 80, 1276-1278 (1974)
Beale, JT: Spectral properties of an acoustic boundary condition. Indiana Univ. Math. J. 25, 895-917 (1976)
Beale, JT: Acoustic scattering from locally reacting surfaces. Indiana Univ. Math. J. 26, 199-222 (1977)
Frota, CL, Cousin, AT, Larkin, NA: Global solvability and asymptotic behaviour of a hyperbolic problem with acoustic boundary conditions. Funkc. Ekvacioj 44(3), 471-485 (2001)
Frota, CL, Goldstein, JA: Some nonlinear wave equations with acoustic boundary conditions. J. Differ. Equ. 164, 92-109 (2000)
Mugnolo, D: Abstract wave equations with acoustic boundary conditions. Math. Nachr. 279, 299-318 (2006)
Vicente, A: Wave equation with acoustic/memory boundary conditions. Bol. Soc. Parana. Mat. 27(1), 29-39 (2009)
Cousin, AT, Frota, CL, Larkin, NA: On a system of Klein-Gordon type equations with acoustic boundary conditions. J. Math. Anal. Appl. 293, 293-309 (2004)
Frota, CL, Larkin, NA: Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. Prog. Nonlinear Differ. Equ. Appl. 66, 297-312 (2005)
Park, JY, Ha, TG: Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions. J. Math. Phys. 50, Article ID 013506 (2009)
Park, JY, Park, SH: Decay rate estimates for wave equations of memory type with acoustic boundary conditions. Nonlinear Anal. 74, 993-998 (2011)
Kang, JR: Exponential decay for a von Karman equation of memory type acoustic boundary conditions. Math. Methods Appl. Sci. 38, 598-608 (2015)
Mustafa, MI, Abusharkh, GA: Plate equations with viscoelastic boundary damping. Indag. Math. 26, 307-323 (2015)
Kang, JR: General stability for a von Karman plate system with memory boundary conditions. Bound. Value Probl. 2015, 167 (2015)
Lions, JL: Quelques methodes de resolution des problemes aux limites non lineaires (1969)
Mustafa, MI, Messaoudi, SA: General stability result for viscoelastic wave equations. J. Math. Phys. 53, 053702 (2012)
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2014R1A1A1003440).
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Kang, JR. General stability for the Kirchhoff-type equation with memory boundary and acoustic boundary conditions. Bound Value Probl 2017, 43 (2017). https://doi.org/10.1186/s13661-017-0774-0
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DOI: https://doi.org/10.1186/s13661-017-0774-0