Abstract
In this paper we consider a von Kármán plate system with memory condition at the boundary. We prove the asymptotic behavior of the corresponding solutions. We establish an explicit and general decay rate result using 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 on some earlier results-exponential or polynomial decay rates.
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1 Introduction
This paper is concerned with the general decay of the solutions to a von Kármán plate system with memory condition at the boundary:
where Ω is a bounded domain of \({\mathbb{R}}^{2}\) with a smooth boundary Γ. Let us denote by \(\nu=(\nu_{1}, \nu_{2} )\) the external unit normal vector on Γ and by \(\tau=(-\nu_{2}, \nu_{1} )\) the corresponding unit tangent vector. The relaxation functions \(g_{1}, g_{2} \in C^{1} (0, \infty)\) are positive and nondecreasing. The von Kármán bracket is given by
Here, \({\mathcal{B}}_{1}\), \({\mathcal{B}}_{2}\) denote the differential operators
and
and the constant \(\mu\in( 0, \frac{1}{2})\) represents Poisson’s ratio.
From the physical point of view, this system describes the transversal displacement u and the Airy-stress function v of a vibrating plate. We know that the memory effect described in integral equations (1.4) and (1.5) can be caused by the interaction with another viscoelastic element. Problems related to equations (1.1)-(1.6) are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics.
The problem of stability of the solutions to a von Kármán system with dissipative effects has been studied by several authors [1–4]. Rivera and Menzala [5] considered the dynamical von Kármán equations for viscoelastic plates under the presence of a long range memory:
The equations describe small vibration of a thin homogeneous, isotropic plate of uniform thickness h. They studied that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation function. Later, Raposo and Santos [6] proved the general decay of the solutions to von Kármán plate model (1.7) under condition on g such as
where ξ is a differential function. This result generalized on the earlier ones in the literature. Kang [7] established an explicit and general decay rate result for von Kármán system (1.7) with nonlinear boundary damping \(h(u_{t})\). Kang improved the results of [6] without imposing any restrictive growth assumption on the damping function h and strongly weakening the usual assumptions on the relaxation function g. Kang [8] showed the exponential decay result of solutions for von Kármán equations (1.7) without the assumption (1.8). She studied that solutions decay exponentially to zero as time goes to infinity in case
for some \(\gamma, \alpha>0\). It is clear then that we are allowing \(g'(t)\) to take negative values. The kernel \(g(t)\) may oscillate. This result improved on the earlier ones concerning the exponential decay. Recently, Kang [9] considered the exponential decay for von Kármán equations (1.7) with acoustic boundary conditions when relaxation function satisfies (1.9). The construction of the Lyapunov function is based on the multiplier method.
On the other hand, Rivera et al. [10] studied the stability of the solutions to a von Kármán system for viscoelastic plates with memory effect in the boundary. They proved that the solution of system (1.1)-(1.6) decays uniformly exponentially or polynomially with the same rate of decay as the relaxation function. Later, Santos and Soufyane [11] improved the decay result of [10]. They assumed that the resolvent kernels satisfy
where \(\gamma_{i} : {\mathbb{R}}^{+} \rightarrow{\mathbb{R}}^{+} \) is a function satisfying the following conditions:
They studied that the energy decays with a rate of decay similar to the relaxation functions, which are not necessarily decaying like polynomial or exponential functions. Motivated by their results, we prove the general decay of the solution for a von Kármán plate system with memory boundary conditions (1.1)-(1.6) for resolvent kernels \(k_{i}\) 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 \(0< r<1\). The proof is based on the multiplier method and makes use of some properties of convex functions including the use of general Young’s inequality and Jensen’s inequality. We establish an explicit decay rate result that allows a wider class of relaxation functions and generalizes previous decay results of [10, 11].
Moreover, there exists a large body of literature regarding viscoelastic problems with the memory term acting at the boundary. Cavalcanti et al. [12] considered the existence and uniform decay rates of solutions to a degenerate system with a memory condition at the boundary. Santos [13] and Santos et al. [14] proved the decay rates for solutions of a Timoshenko system and a nonlinear wave equation of Kirchhoff type with a memory condition at the boundary, respectively. Park and Kang [15] investigated the asymptotic behavior of the solutions of a multi-valued hyperbolic differential inclusion with a boundary condition of memory type. Precisely, denoting by k the resolvent kernel of \(- {g' / g(0)}\), they proved that the energy of the solution decays exponentially (polynomially) to zero provided k decays exponentially (polynomially) to zero. Messaoudi and Soufyane [16], Mustafa and Messaoudi [17] and Kang [18, 19] obtained the general stability for wave equation, the Timoshenko system and Kirchhoff plates with viscoelastic boundary conditions under condition k satisfying (1.10), respectively. Mustafa and Abusharkh [20] proved the general decay for plate equations with viscoelastic boundary damping when resolvent kernels \(k_{i}\) satisfy (1.11).
Besides, Mustafa and Messaoudi [21, 22] investigated the general stability result of viscoelastic equation for relaxation function g satisfying (1.11). These conditions on H are weaker than those imposed in [23].
The paper is organized as follows. In Section 2 we present some notations and material needed for our work. In Section 3 we prove the general decay of the solutions to the von Kármán plate system with memory condition at the boundary.
2 Preliminaries
In this section, we present some material needed in the proof of our main result. Throughout this paper we define
For a Banach space \(X, \|\cdot\|_{X}\) denotes the norm of X. For simplicity, we denote \(\|\cdot\|_{L^{2}(\Omega)}\) and \(\|\cdot\|_{L^{2} (\Gamma)}\) by \(\|\cdot\|\) and \(\|\cdot\|_{\Gamma}\), respectively. A simple calculation, based on the integration by parts formula, yields
where the bilinear symmetric form \(a(u, v)\) is given by
We know that \(\sqrt{a(u,u)}\) is equivalent to \(H^{2} (\Omega)\), that is,
where \(c_{0}\) and \(\tilde{c}_{0}\) are generic positive constants. This and the Sobolev imbedding theorem imply that for some positive constants \(C_{p}\) and \(C_{s}\),
We assume that there exists \(x_{0} \in{\mathbb{R}}^{2}\) such that
We define \(m(x) = x - x_{0}\) and \(R=\max_{x\in\Omega}|m(x)| \).
The following identity will be used later.
Lemma 2.1
([24])
For every \(u\in H^{4} (\Omega)\), we have
Now, we introduce the relative results of the Airy stress function and von Karman bracket \([\cdot, \cdot]\).
Lemma 2.2
([25])
Let u, w be functions in \(H^{2} (\Omega)\) and v in \(H^{2}_{0}(\Omega)\), where Ω is an open bounded and connected set of \({\mathbb{R}}^{2}\) with regular boundary. Then
Now, we use the boundary conditions (1.4) and (1.5) to estimate the terms \({\mathcal{B}}_{1} u\) and \({\mathcal{B}}_{2} u\). Denoting by
the convolution product operator and differentiating equations (1.4) and (1.5), we arrive at the following Volterra equation:
Applying Volterra’s inverse operator, we have
where the resolvent kernels \(k_{i}\) (\(i=1, 2\)) satisfy
Assuming throughout the paper that \(u_{0} \equiv0\) on \(\Gamma\times (0, \infty)\) and defining \(\tau_{1} = {\frac{1}{{g_{1}(0)}}}\) and \(\tau_{2}=\frac{1}{g_{2}(0)}\), we can rewrite \({\mathcal{B}}_{1} u\) and \({\mathcal{B}}_{2} u\) as
Hence, we use conditions (2.4) and (2.5) instead of the boundary conditions (1.4) and (1.5).
Let us define
By differentiating the term \(g \,\square\, v\), we obtain the following lemma for the important property of the convolution product operator.
Lemma 2.3
For \(g, v \in C^{1} ( [ 0, \infty): {\mathbb{R}} )\), we have
We consider the following assumptions on \(k_{1}\) and \(k_{2}\).
(A1) The resolvent kernels \(k_{i} : {\mathbb{R}}_{+} \rightarrow{\mathbb{R}}_{+} \) (\(i=1, 2\)) are twice differentiable functions such that
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
The energy of system (1.1)-(1.6) is given by
The well-posedness of von Kármán system plates with boundary conditions of memory type is given by the following theorem.
Theorem 2.1
([10])
Let \(k_{i} \in C^{2} ( {\mathbb{R}}_{+} )\) be such that \(k_{i}, - k_{i}' , k_{i}'' \geq0 \) for \(i=1, 2\). If \(( u_{0} , u_{1} ) \in H^{2}(\Omega) \cap L^{2} (\Omega)\), then there exists a unique weak solution for system (1.1)-(1.6). Moreover, if \((u_{0}, u_{1})\in (H^{4} (\Omega)\cap H^{2} (\Omega)) \times H^{2} (\Omega)\), then the solution of (1.1)-(1.6) has the following regularity:
We are now ready to state our main result.
Theorem 2.2
Assume that (A1) holds. Suppose 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
Then there exist positive constants \(c_{1}\), \(c_{2}\), \(c_{3}\) and \(\epsilon_{0}\) such that the solution of (1.1)-(1.6) satisfies
where
Moreover, for some choice of D, if \({\int_{0}^{1} H_{1} (t)\,dt } < +\infty\), then we obtain
where
In particular, (2.9) is valid for the special case \(H(t) = c t^{p}\) for \(1\leq p< \frac{3}{2}\).
Remark 2.1
(1) From (A1), we conclude that \({\lim_{t \rightarrow+\infty} (-k_{i}' (t)) =0}\) for \(i=1, 2\). This implies that \({\lim_{t \rightarrow+\infty} k_{i}'' (t)}\) cannot be equal to a positive number, and so it is natural to assume that \({\lim_{t \rightarrow+\infty} k_{i}'' (t) =0}\). Therefore, there is \(t_{0} >0\) large enough such that \(k'_{i}(t_{0}) <0\) and
From H is a positive continuous function, we obtain
for some positive constants \(d_{1}\) and \(d_{2}\). Since \(k_{i}'\) is nondecreasing, \(k_{i}'(0)<0\) and \(k_{i}'(t_{0} ) <0\), we have
Hence, by (2.6), (2.12) and (2.13),
which gives
for some positive constant \(d_{3}\).
(2) By using the properties of H, we can prove that the function \(H_{1}\) is strictly decreasing and convex on \((0, 1]\), with \({\lim_{t\rightarrow0}H_{1} (t) = +\infty.}\) Thus, Theorem 2.2 ensures
Remark 2.2
The well-known Jensen’s inequality will be of essential use in establishing our main result. 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 General decay
In this section, we study the asymptotic behavior of the solutions for system (1.1)-(1.6). To prove the decay property, we first obtain the dissipative property of system (1.1)-(1.6). Multiplying equation (1.1) by \(u_{t}\) and integrating by parts over Ω, we get
From the boundary conditions (2.4) and (2.5) and Lemma 2.3, we have
Let us introduce the following functional:
The following lemma plays an important role in the construction of the Lyapunov functional.
Lemma 3.1
There exists \(C>0\) such that
Proof
Differentiating \(\Phi(t)\) and using (1.1), (2.1) and Lemma 2.1, we obtain
According to Lemma 2.2, we have
Using the trace theorem, (2.2), (2.3) and Young’s inequality, we get, for \(\epsilon_{1} >0\),
where \(\epsilon_{1}\) is a positive constant. From (3.3)-(3.5), we obtain
Note that
From the above inequality, the boundary conditions (2.4) and (2.5) can be written as
Substituting (3.7) and (3.8) into (3.6) and choosing \(\epsilon_{1}\) small enough, we have estimate (3.2). □
Let us consider the following binary operator:
Then, applying Hölder’s inequality for \(0 \leq\alpha\leq1\), we get
Proof of Theorem 2.2
Let us introduce the Lyapunov functional
with \(N >0\).
Using (3.1), (3.2) and (3.9) with \(\alpha= \frac{1}{2}\), we obtain
Taking N large, for some positive constant β,
which, using trace theory and the fact that \({\lim_{t\rightarrow\infty}} k_{i}(t) =0\) for \(i=1, 2\), yields, for large \(t_{0}\),
On the other hand, it is not difficult to see that \(L(t)\) satisfies
for some positive constants \(q_{0}\), \(q_{1}\). Using (2.14), (3.1) and (3.10), we have
where \(\tau=\min\{ \tau_{1}, \tau_{2} \}\). We take \({\mathcal{L}}(t) = L(t) +\frac{2C}{d_{3} \tau}E(t)\), which is equivalent to \(E(t)\), and use (3.12) to get
(A) The general case: This case is obtained on account of the ideas presented in [20, 22] as follows. Let \(H_{0}^{*}\) be the convex conjugate of \(H_{0}\) in the sense of Young (see [26]); then
and \(H_{0}^{*}\) satisfies the following Young’s inequality:
We define \(\eta(t)\) and \(\xi(t)\) by
where \(H_{0}\) is such that (2.7) is satisfied. From (2.7), (3.1) and trace theory, and choosing \(t_{0}\) even larger if needed, we find that \(\eta(t)\) and \(\xi(t)\) satisfy
Besides, we define \(\kappa(t)\) and \(\chi(t)\) by
By (2.6) and the properties of \(H_{0}\) and D, we obtain
for some positive constant \(\alpha_{0}\). Then, using (2.11), (3.1) and (3.17) and choosing \(t_{0}\) even larger, we can see that \(\kappa(t)\) satisfies, for all \(t\geq t_{0}\),
Similarly, we deduce that \(\chi(t) \leq\min\{ r, H(r), H_{0} (r)\}\). Since \(H_{0}\) is strictly convex on \((0, r]\) and \(H_{0} (0)=0\), then
provided \(0\leq \lambda\leq1\) and \(x\in(0, r]\). Using (3.16), (3.19) and Jensen’s inequality (2.15), we have
This implies that
Similarly, we get
From (3.13), (3.20) and (3.21) we deduce that
For \(\epsilon_{0} < r\) and \(d_{0} > 0\), we define the functional
which satisfies
for some \(\alpha_{1}, \alpha_{2}>0\). Also, by \(\epsilon_{0} < r\), \({ E}' \leq 0\), we get \(\epsilon_{0} \frac{{ E}(t)}{{ E}(0)}< r\). Using (3.1), (3.14), (3.15), (3.18) and (3.22) and the fact that \({ E}' \leq0\), \(H_{0} >0\), \(H_{0}' > 0\) and \(H_{0}''> 0\) on \((0, r]\), we obtain
Therefore, with a suitable choice of \(\epsilon_{0}\) and \(d_{0}\), we have, for all \(t \geq t_{0}\),
where \(\alpha_{3}>0\) and \(H_{2} (t) = t H_{0}' (\epsilon_{0} t)\). By the strict convexity of \(H_{0}\) on \((0, r]\), we see that \(H_{2} (t) >0\), and \(H_{2} ' (t)=H_{0}'(\epsilon_{0} t)+\epsilon_{0} t H_{0}'' ( \epsilon_{0} t)>0\) on \((0, 1]\). We take
which is clearly equivalent to \({E}(t)\). Using (3.23), (3.24) and \(H_{2}'>0\), we get
where \(k_{0} = \frac{ \alpha_{1} \alpha_{3}}{{ E}(0)} >0 \). Then, using the properties of \(H_{2}\), the fact that \(H_{1}\) is a strictly decreasing function on \((0, 1]\) and \(\lim_{t\rightarrow0} H_{1} (t)= +\infty\), we obtain, for some \(k_{1}, k_{2}>0\),
where \(H_{1} (t) =\int_{t}^{1} \frac{1}{H_{2} (s)}\,ds\). Using (3.23) and (3.25), we have (2.8). Moreover, if \(\int_{0}^{t} H_{1} (t)\,dt < +\infty\), then \(\int_{0}^{+\infty} H_{1} ^{-1} (t)\,dt < +\infty\). From (2.8) we get \(\int_{0}^{+\infty} E(t)\,dt < \infty\) and
Similarly, we define, for large \(t_{0}\),
and
From (2.6), the strict convexity of H and Jensen’s inequality (2.15), we obtain
Therefore, we deduce that
Similarly, we have
Thus, (3.13) becomes
Hence, repeating the same procedures, we see that for some \(c_{1}\), \(c_{2}\) and \(c_{3} >0\),
where \(G(t) = \int_{t}^{1} \frac{1}{sH'(\epsilon_{0} s)}\,ds \).
(B) The special case \(H(t) =ct^{p} \) and \(1\leq p< \frac{3}{2}\):
Case 1. \(p=1\): Using (2.6) and (3.1), estimate (3.13) yields, for all \(t \geq t_{0} \),
which gives
From (3.11) we know that \({\mathcal{L}}+ \frac{2C}{c\tau} { E} \sim E\). Hence, we easily get
where
Case 2. \(1< p<\frac{3}{2}\): We can see that
for any \(\delta_{0} <2-p\). Using (3.1) and (3.26) and choosing \(t_{0}\) even larger if needed, we conclude that for all \(t \geq t_{0}\),
and
From Hölder’s inequality, Jensen’s inequality, (2.6), (3.1) and (3.27), we get
Similarly, we obtain
Therefore, using (3.28) and (3.29), we see that (3.13) yields, for \(\delta_{0} =\frac{1}{2}\),
Multiplying (3.30) by \({ E}^{\gamma}(t)\), with \(\gamma=2p-2\), and using (3.1) and Young’s inequality, with \(\frac{1}{\gamma+1}+\frac{\gamma}{\gamma+1}=1\), we have
Choosing \(\varepsilon<\beta\), we get, for some \(C_{1}>0\),
where \(L_{0} ={\mathcal{L} }{ E}^{\gamma}+C_{\varepsilon}{ E} \sim E \). Then we conclude that
Since \(p<\frac{3}{2}\) and using (3.31), we see that
Using this fact, we have
Then, by (2.6), (3.1), (3.32), (3.33) and Hölder’s inequality, estimate (3.13) gives
Now, we multiply (3.34) by \({ E}^{\gamma}(t)\) with \(\gamma=p-1\). Hence, repeating the above steps, we find that
where
□
Example
We give an example to explain the energy decay rates given by Theorem 2.2.
(1) If
for \(q>3\) and \(a>1\), then \(k_{i}''(t) =H(-k_{i}'(t))\), where
Because
then the function H satisfies hypothesis (A1) on the interval \((0, r]\) for any \(0< r<\frac{1+q-\sqrt{q^{2}-1}}{2aq}\). By taking \(D(t)=t^{\alpha}\), (2.7) is satisfied for any \(\alpha>\frac{q}{q-1}\). Hence, an explicit rate of decay can be obtained by Theorem 2.2. The function \(H_{0} (t) = H(t^{\alpha})\) has derivative
Thus,
Let \(\frac{1}{(\epsilon_{0} s)^{\alpha}} =u\), then we have
Using the fact that the function \(f(u) = (u-a)^{\frac{1}{q}}\) is increasing on \((a, +\infty)\) and \((u-a)^{\frac{1}{q}} < u^{\frac{1}{q}}\) and taking \(\epsilon_{0} < a^{-\frac{1}{\alpha}}\), we get
Next, we find that if \(\alpha< \frac{2q}{1+q}\),
Taking \(\frac{1}{\epsilon_{0} s} =v \) and \(\epsilon_{0} < a^{-1}\), we obtain
Therefore,
Hence we can use (2.9) to deduce that the energy decays
where \(\tilde{c}_{i}\) (\(i=1, 2 , 3\)) are constants.
(2) As in [20], let \(0< q<1\)
then \(k_{i}''(t)=H(-k_{i}'(t))\), where, for \(t\in(0, r]\), \(r<1\), \(H(t)= \frac {qt }{[\ln(1/t)]^{\frac{1}{q}-1}}\). Hence
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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 a von Kármán plate system with memory boundary conditions. Bound Value Probl 2015, 167 (2015). https://doi.org/10.1186/s13661-015-0431-4
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DOI: https://doi.org/10.1186/s13661-015-0431-4