Abstract
A nonlinear viscoelastic Kirchhoff plate equation with a time delay term in the internal feedback is considered. Under suitable assumptions, we establish the general rates of energy decay of the initial and boundary value problem by using the energy perturbation method.
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1 Introduction
In this paper, we are concerned with the following nonlinear viscoelastic Kirchhoff plate equation with a time delay term in the internal feedback:
where \(\Omega\subseteq {\mathbb{R}}^{n} \) (\(n\geq1\)) is a bounded domain with smooth boundary ∂Ω. The function \(u=u(x,t)\) is the transverse displacement of a plate filament, and \(\sigma(t)\) and \(g(t)\) are positive functions defined on \(\mathbb{R}^{+}\). \(\mu_{1}\), \(\mu_{2}\) are positive constants and \(\tau>0\) represents the time delay.
To equation (1.1), we add the following initial conditions:
and the support boundary conditions
In 1950, Woinowsky-Krieger [1] introduced the one-dimensional nonlinear equation of vibration of beams
where L is the length of the beam and \(\alpha,\beta,\gamma\) are positive physical constants. Since then many mathematicians studied the related model in one dimension and higher dimensions. The main results are mainly concerned with global existence, stability, and long-time dynamics, and many results may be found in the literature. It has been stabilized by means of different controls, for example, internal damping, boundary controls, dynamic boundary conditions, distributed damping and heat damping, and so on. See, for example, Brito [2], Cavalcanti et al. [3–5], Jorge Silva and Ma [6], Ma [7], Ma and Narciso [8], Oliveira and Lima [9], Park [10], Patcheu [11], Munõz Rivera [12, 13], Yang [14, 15], and the references therein. We would like here to mention the work of Andrade et al. [16]. In this paper the authors studied a viscoelastic plate equation with p-Laplacian and memory terms with strong damping,
and proved the existence of weak solutions by using Faedo-Galerkin approximations to the IBVP of (1.4). In addition, they obtained the uniqueness of strong solutions and the exponential stability of solutions to (1.4) under some suitable conditions on the memory kernel g and a forcing term f. For \(\sigma(t)>0\), Messaoudi [17] considered the following viscoelastic wave equation:
Under some assumptions on the relaxation function g and the potential σ, the author established a general decay property which depends on the behavior of σ and g. Jorge Silva et al. [18] studied the following viscoelastic Kirchhoff plate equation:
and they mainly proved the global well-posedness of the solution for \(\sigma(t)=1\) and similarly the result holds for \(\sigma(t)=0\). Moreover, the authors established the general rates of energy decay of the system for \(\sigma\in[0,\infty)\). For more results on viscoelastic equations, we can refer to Berrimi and Messaoudi [19], Messaoudi [20], Messaoudi and Tartar [21, 22], Tatar [23], and the references therein.
In recent years, there has been published much work concerning the wave equation with time delay effects and the delay effects often appear in many practical problems. In Nicaise and Pignotti [24], the authors studied a wave equation with time delay,
and established stability results under the assumption \(0<\mu_{2}<\mu_{1}\). In [25], Kirane and Said-Houari studied a viscoelastic wave equation with a delay term in internal feedbacks, and they proved the global well-posedness of the IBVP to the equation by using some suitable assumptions on the relaxation function and some restriction on the parameters \(\mu_{1}\) and \(\mu_{2}\). Furthermore, under the assumption \(\mu_{2}\leq\mu_{1}\), they obtained a general decay result of the total energy to the system. Dai and Yang [26] improved the results in [25] under weaker conditions. For the plate equation with time delay term, Park [27] considered
which can be regarded as an extensive weak viscoelastic plate equation with a linear time delay term. The author obtained a general decay result of energy by using suitable energy and Lyapunov functionals. In [28], one of the present authors investigated an extensible plate equation with a weak viscoelastic term and a time delay term in the internal feedback,
and established the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations and some energy estimates. Moreover, the author proved a general rate result of energy decay when the weight of the delay is less than the weight of the damping. Recently, Yang [29] studied a viscoelastic plate equation with a linear time delay term
The author obtained the global well-posedness of the IBVP to the equation and established the decay property of energy for either \(0<|\mu_{2}|<\mu_{1}\) or \(\mu_{1}=0\), \(0<|\mu_{2}|<a\), and \(\zeta_{2}>\zeta_{0}\), but one needs more assumptions on the kernel g. For more some results concerning the different boundary conditions under an appropriate assumption between \(\mu_{1}\) and \(\mu_{2}\), one can refer to Datko et al. [30], Kafini et al. [31], Nicaise and Pignotti [32], Nicaise et al. [33], Nicaise and Valein [34], and the references therein.
Equation (1.1) is a Kirchhoff plate equation with a memory term and a nonlinear time delay term in the internal feedback. To the best of our knowledge, the general rate of energy decay for system (1.1)-(1.3) were not previously considered. So the main objective of the present work is to establish the stability of initial boundary value problem (1.1)-(1.3).
The outline of this paper is as follows. In Section 2, we give some preparations for our consideration and our main results. In Section 3, we establish the general decay result of the energy by using energy perturbation method.
2 Assumptions and main results
We first introduce the following Hilbert spaces:
with norms
respectively. The notation \(\|\cdot\|_{p}\) denotes the \(L^{p}\)-norm, and \((\cdot,\cdot)\) is the \(L^{2}\)-inner product. In particular, we write \(\|\cdot\|\) instead of \(\|\cdot\|_{2}\) when \(p=2\). The constants \(\lambda_{0},\lambda_{1},\lambda_{2},\lambda>0\) represent the embedding constants
for \(u\in V_{2}\).
For the relaxation function g and the potential σ, we assume
(A1) \(g,\sigma:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) are nonincreasing differentiable functions satisfying
with \(l=1-l_{0}\), and there exists a nonincreasing differentiable function \(\zeta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) satisfying
(A2) The constant m satisfies
(A3) The function \(F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is a \(C^{1}\)-vector field given by \(F=(F_{1},\ldots,F_{n})\) satisfying for every \(j=1,2,\ldots,n\),
where \(k_{j}\) are positive constants and the constants \(p_{j}\) satisfy
Moreover, the function F is a conservative vector field with \(F=\nabla f\), where \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}\) is a real valued function satisfying
where \(\alpha\in[0,\mu)\) with \(\mu=\lambda_{2}\frac{1-2\sigma(t)\int^{t}_{0}g(s)\,ds}{2l}\).
The vector field F satisfying a condition like (2.4) possesses an interesting property. One can find the detailed proof in [18].
Remark 2.1
Let \(F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is a \(C^{1}\)-vector field given by \(F=(F_{1},\ldots,F_{n})\). If there exist positive constants \(k_{1},\ldots,k_{n}\) and \(q_{1},\ldots,q_{n}\) such that, for every \(j=1,\ldots,n\),
Then there exists a positive constant \(K=K(k_{j},q_{j},n), j=1,\ldots,n\), such that, for all \(x,y\in\mathbb{R}^{n}\),
In particular, we have
Now we give some estimates related to the convolution operator. By direct calculations, we shall see below that
where
Motivated by [32, 34], we introduce the following new dependent variable to deal with the delay feedback term:
which gives us
Thus, problem (1.1)-(1.3) is equivalent to
where \(x\in\Omega,\rho\in(0,1)\) and \(t>0\), and the initial and boundary conditions are
Let ξ be a positive constant satisfying
Now we define the weak solutions of (1.1)-(1.3): for given initial data \((u_{0},u_{1})\in V_{2}\times V_{0} \), we say that a function \(U=(u,u_{t})\in C(\mathbb{R}^{+},V_{2}\times V_{0})\) is a weak solution to the problem (1.1)-(1.3) if \(U(0)=(u_{0},u_{1})\) and
for all \(\omega\in V_{2}\).
The following theorem is concerned with the global well-posedness of problem (2.12)-(2.13). By using the classical Faedo-Galerkin method, see, e.g., [16, 18, 28, 35], we can prove the theorem, and we omit the proof here.
Theorem 2.1
Let \(\mu_{2}\leq m\mu_{1}\), and assume the assumptions (2.1)-(2.6) hold. If the initial data \((u_{0},u_{1})\in(V_{2}\times V_{0})\), \(f_{0}\in L^{2}(\Omega\times(0,1))\), then problem (2.12)-(2.13) has a unique weak solution \((u,u_{t})\in C(0,T;V_{2}\times V_{0})\) such that, for any \(T>0\),
We introduce the modified energy functional to problem (2.12)-(2.13) by
Our main result is the general decay rate of the energy, which is given by the following theorem.
Theorem 2.2
Let \(\mu_{2}< m\mu_{1}\), and assume the assumptions (2.1)-(2.6) hold. Let \((u,u_{t})\) be the weak solutions of problem (2.12)-(2.13) with the initial data \((u_{0},u_{1})\in(V_{2}\times V_{0})\), \(f_{0}\in L^{2}(\Omega\times(0,1))\). Then there exist two constants \(\beta>0\) and \(\gamma>0\) such that the energy \(E(t)\) defined by (2.15) satisfies
Remark 2.2
Generally speaking, the energy of problem (2.12)-(2.13) is usually defined by
From Theorem 2.2, we can also get the decay
Indeed, by (2.15) and (2.1), we have
which, together with (2.16), implies (2.17) with \(\beta '=\frac{4\beta}{l+3}\).
3 General decay rate
In this section, we shall establish the general decay property of the solution for problem (2.12)-(2.13) in the case \(\mu_{2}< m\mu_{1}\). For this purpose we define
where \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are positive constants and
To prove Theorem 2.2, we need the following technical lemmas.
Lemma 3.1
Under the assumptions in Theorem 2.2, the modified energy functional defined by (2.15) satisfies there exist two positive constants \(c_{1}\) and \(c_{2}\) such that, for any \(t\geq0\),
Proof
First the direct calculation yields
Multiplying the first equation in (2.12) by \(u_{t}\), integrating the result over Ω, and using integration by parts, (2.8) and (3.5), we can obtain
Multiplying the second equation in (2.12) by ξz and integrating the result over \(\Omega\times(0,1)\), we have
By using Young’s inequality, we get
which, together with (3.6)-(3.7), gives us
By using condition (2.14), we get
which implies the desired inequality (3.4). The proof is now complete. □
Lemma 3.2
Under the assumptions in Theorem 2.2, for the functional \(\Phi(t)\) defined in (3.2) there exists a positive constant \(c_{3}\) such that, for any \(t\geq0\),
where \(C_{\varepsilon}>0\) is a constant depending for any \(\varepsilon>0\).
Proof
By using the first equation of (2.12), we obtain
Using Young’s inequality, the embedding theorem, and (2.15), we know that, for any \(\varepsilon>0\),
where the constant \(C>0\) is the embedding constant.
Similarly we get
We infer from (2.6) that
Combining (2.11) and (3.10)-(3.12) with (3.9), we can get
Due to (2.1) and (2.6) and choosing \(\varepsilon>0\) small enough, we know that
which, together with (3.13), give us (3.8). The proof is hence complete. □
Lemma 3.3
Under the assumptions in Theorem 2.2, and for any \(\delta>0\), there exists a positive constant \(C_{\delta}\) such that the functional \(\Psi(t)\) defined in (2.3) satisfies
where
Proof
The straightforward computation implies that
By using Hölder’s inequality, Young’s inequality, and the embedding theorem, we can infer that, for any \(\delta>0\),
Now we estimate the term \(\int_{\Omega}F(\nabla u)\int^{t}_{0}g(t-s)(\nabla u(t)-\nabla u(s))\,ds\,dx\) by using the method in [18]. Noting (2.5) and the embedding \(V_{2}\hookrightarrow W^{1,p_{j}+1}_{0}(\Omega)\), \(j=1,\ldots,n\), we know that there exist positive constants \(\mu_{p_{1}},\ldots,\mu_{p_{n}}\) satisfying
Using (2.7) with \(F(0)=0\), Hölder’s inequality, Young’s inequality, and the embedding theorem, we can conclude that, for any \(\delta>0\),
where
and
It follows from (3.21), Hölder’s inequality, and Young’s inequality that
Inserting (3.16)-(3.20) and (3.22) into (3.15), we obtain (3.14). The proof is therefore complete. □
Lemma 3.4
For \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) small enough, there exist two positive constants \(\beta_{1}\) and \(\beta_{2}\) such that
Proof
Using Hölder’s inequality, Young’s inequality, and Poincaré’s inequality, we can easily get
which, choosing \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) small enough, implies (3.23). The proof is complete. □
Proof of Theorem 2.2
Combining (3.4), (3.8), and (3.14) with assumption (A1), we can obtain
By using the Young inequality and the Poincaré inequality, we shall see that
For any fixed \(t_{0}>0\), we know that, for any \(t\geq t_{0}\),
which, along with (3.24)-(3.25), implies for any \(t\geq t_{0}\),
At this point we first choose \(0<\delta<\min\{\frac{g_{0}}{2},\frac{c_{3}g_{0}}{4[1+(1-l)^{2}]}\}\), and we get
For any fixed \(\delta>0\), we take \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) satisfying
so small that
We at last choose \(\varepsilon_{1}>0\) and \(\varepsilon_{2}>0\) small enough for (3.23) and (3.27) to remain valid, and further,
From this it follows that, for positive constants \(\eta_{1}\), \(\eta_{2}\), and \(\eta_{3}\),
Since \(\lim_{t\rightarrow\infty}\frac{\sigma'(t)}{\sigma(t)}=0\), we choose \(t_{1}\geq t_{0}\) and use (2.16) to get
where \(\eta_{4}\) and \(\eta_{5}\) are positive constants.
Multiplying (3.29) by \(\zeta(t)\) and using (3.4), we obtain
which, combining with (2.16), gives us for any \(t\geq t_{1}\),
Since \(\lim_{t\rightarrow\infty}\frac{\sigma'(t)}{\zeta(t)\sigma(t)}=0\), we can choose \(t_{2}\geq t_{1}\) so that
Let \(\mathcal{E}(t)=\zeta(t)\mathcal {L}(t)+2\eta_{5}E(t)\), then it is easy to see that \(\mathcal{E}(t)\) is equivalent to the modified energy \(E(t)\) by using (3.23), that is, there exist two positive constants \(\beta_{3}\) and \(\beta_{4}\) such that
which, together with (3.31) and using \(\zeta'(t)\leq0\), shows that there exists a positive constant \(\gamma_{1}>0\) such that, for any \(t\geq t_{2}\),
Integrating (3.33) over \((t_{2},t)\) with respect to t, we get for any \(t\geq t_{2}\),
which, using (3.32), implies for any \(t\geq t_{2}\),
Therefore (2.16) follows by renaming the constants, and by the continuity and boundedness of \(E(t)\). The proof is hence complete. □
Remark 3.5
We illustrate several rates of energy decay through the following examples, some of which can be found in [17, 23].
Example 1
If g decays exponentially, i.e., \(\zeta(t)=a\), and \(\sigma(t)=\frac{b}{1+t}\), then (2.16) gives us
Example 2
If g decays exponentially, i.e., \(\zeta(t)=a\), and \(\sigma(t)=b\), then (2.16) gives us
Example 3
When \(g(t)=ae^{-b(1+t)^{\alpha}}\) and \(\sigma(t)=\frac{1}{1+t}\) for \(a,b>0\) and \(0<\alpha<1\), then \(\zeta(t)=b\alpha(1+t)^{\alpha-1}\) satisfies (2.1)-(2.2). Estimate (2.16) takes the form
Example 4
If \(g(t)=a\exp(-b\ln^{\alpha}(1+t))\) and \(\sigma(t)=\frac{1}{\ln(1+t)}\) for \(a,b>0\) and \(\alpha>1\), we know that \(\zeta(t)=\frac{b\alpha \ln^{\alpha-1}(1+t)}{1+t}\) satisfies (2.1)-(2.2). Estimate (2.16) takes the form
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Acknowledgements
This work was support by the NSFC (No. 11526164) and the Fundamental Research Funds for the Central Universities with grant number JBK160122.
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Feng, B., Li, H. Energy decay for a viscoelastic Kirchhoff plate equation with a delay term. Bound Value Probl 2016, 174 (2016). https://doi.org/10.1186/s13661-016-0682-8
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DOI: https://doi.org/10.1186/s13661-016-0682-8