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Global existence and stability results for a nonlinear Timoshenko system of thermoelasticity of type III with delay
Boundary Value Problems volume 2018, Article number: 65 (2018)
Abstract
In this paper, we consider a nonlinear thermoelastic system of Timoshenko type with delay. It is known that an arbitrarily small delay may be the source of instability. The delay term works on the second equation which describes the motion of a rotation angle. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation. Our results show that the damping effect is strong enough to uniformly stabilize the system even in the presence of time delay under suitable conditions by using perturbed energy functional technique and improve the related results.
1 Introduction
In this paper, we consider a nonlinear Timoshenko-type system of thermoelasticity of type III with delay:
in which \(\rho_{1}\), \(\rho_{2}\), \(\rho_{3}\), K, b, k, β, γ, δ, \(\mu_{1}\), \(\mu _{2}\), τ are positive constants. In this system, \(\mu_{1}\psi_{t}\) represents a frictional damping, \(\mu _{2}\psi_{t}(x,t-\tau)\) represents a delay term, and \(f(\psi)\) is a forcing term. We impose the following initial and boundary conditions:
We start our literature review with the pioneer work of Messaoudi and Said-Houari [1]. The authors considered the system as follows:
They obtained an exponential decay result under equal wave speeds (\(\frac{K}{\rho_{1}}=\frac{b}{\rho_{2}}\)). Later Messaoudi and Fareh [2] studied the case of nonequal speeds (\(\frac{K}{\rho_{1}}\neq\frac {b}{\rho_{2}}\)) and established a polynomial decay result. In [3], the author consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound as follows:
They established general decay results for the cases of \(\mu=0\) and \(\mu\neq0\) with the constant \(\mu=(\tau-\frac{ \rho_{1}}{K\rho _{3}})(\frac{\rho_{2} }{b}-\frac{\rho_{1} }{K})-\frac{\tau\delta^{2}\rho _{1}}{bK\rho_{3}}\).
On the other hand, Timoshenko systems with delay term have attracted extensive attention, and the increasing complexity of their types makes research more significant. An arbitrarily small delay may be the source of instability, see [4, 5]. Racke [6] considered the following coupled system of linear thermoelastic equations with constant delays \(\tau_{1}\) and \(\tau_{2}\):
He obtained that the solution of problem (1.4) is instable with any delay \(\tau_{1}>0\) or \(\tau_{2}>0\). In recent years constant delay τ has been extended to the time-varying function \(\tau(t)\) in the thermoelastic equations, see [7]. Also, in [8] Nicaise and Pignotti studied the initial-boundary value problem of wave equation with boundary distributed delay as follows:
They proved an exponential stability result for system (1.5) with the condition
and when the boundary distributed delay term in the above system is replaced by the internal feedback \(\int_{\tau_{1}}^{\tau_{2}}a(x)\mu (s)u_{t}(t-s)\,ds\) with \(a(x)\) satisfying some suitable conditions. They also obtained that the energy of solution is exponentially decaying to zero under the condition
We refer the reader to [9–14] for more analogous results. Kafini et al. [15], considered a one-dimensional Timoshenko-type system
They proved the well-posedness of system (1.6) and established an exponential decay result under the condition \(\frac{K}{\rho_{1}}=\frac {b}{\rho_{2}}\) and a polynomial decay result under the condition \(\frac{K}{\rho_{1}}\neq \frac{b}{\rho_{2}}\). For a Timoshenko system with time delay and forcing term at the same time
Feng and Pelicer [16] obtained an exponential stability under equal wave speeds. In the present paper, when \(\frac{K}{\rho_{1}}=\frac{b}{\rho_{2}}\), we extend their result to nonlinear Timoshenko system of thermoelasticity of type III by using the perturbed energy functional technique as well, and when \(\frac{K}{\rho_{1}}\neq\frac{b}{\rho_{2}}\), we achieve a polynomial decay estimate.
This paper is organized as follows. In Sect. 2, we present some assumptions and preliminary works. In Sect. 3, we establish the well-posedness of system (1.1)–(1.2) by using semigroup theory in [15, 16]. In Sect. 4, we prove the decay results in two cases by using energy methods.
2 Preliminaries
In this section, we present some materials needed for our main results. For simplicity of notations, hereafter we denote by \(\| \cdot\|_{q}\) the Lebesgue space \(L^{q}(\Omega)\) norm, and by \(\| \cdot\|\) the Lebesgue space \(L^{2}(\Omega)\) norm.
Assumption 2.1
Assume that \(f:\mathbb{R}\rightarrow\mathbb{R}\) with \(f(0)=0\) satisfies
where \(k_{0}>0\), \(\varsigma\geq1\) are constants such that
In addition we assume that
in which \(\hat{f}(\psi): =\int_{0}^{\psi}f(s)\,ds\).
In order to deal with the delay term, we define the following new variable:
Thus we have
Then system (1.1)–(1.2) is transformed to
In order to use the Poincaré inequality for θ, as in [15], we introduce
Then by (2.4)3 we have
After a simple substitution, we see that \((\phi,\psi,\bar{\theta},z)\) satisfies (2.4). From now on, we work with θ̄ but write θ for convenience.
3 Well-posedness result
In this section, we shall investigate the well-posedness of problem (2.1) with semigroup theory, we start with the vector function \(\mathcal{U}(t)=(\phi,\varphi,\psi,u,\theta ,v,z)^{T}\), where \(\varphi=\phi_{t}\), \(u=\psi_{t}\), and \(v=\theta_{t}\). We introduce as in [15]
Then we define the energy space by
equipped with the following inner product:
in which ξ is a positive constant satisfying
Thus system (2.4) can be re-written as
where the operators \(\mathcal{A}\) and \(\mathcal{F}\) are defined by
with
and the initial value \((\theta_{0},\theta_{1})\) satisfies
By using the same methods as those in [15] and in [16], we can obtain the following Lemmas 3.1 and 3.2, respectively. We omit the proof.
Lemma 3.1
The operator \(\mathcal{A}\) defined in (3.3) is the infinitesimal generator of a \(C_{0}\)-semigroup in \(\mathcal{H}\).
Lemma 3.2
The operator \(\mathcal{F}\) is locally Lipschitz in \(\mathcal{H}\).
According to Pazy [17], Chap. 6, we can obtain the existing results as follows. We omit the proof.
Theorem 3.3
Suppose that Assumption 2.1 holds and \(\mu_{2}\leq\mu_{1} \). For any initial value \(\mathcal{U}_{0}\in\mathcal{H}\), system (3.3) admits a unique solution \(\mathcal{U}\in C(0,\infty;\mathcal{H})\). Moreover, if \(\mathcal{U}_{0}\in\mathcal{D(A)}\), system (3.3) admits a unique solution \(\mathcal{U}\in C(0,\infty;\mathcal {D(A)})\cap C^{1}(0,\infty;\mathcal{H})\).
We introduce the first order energy of problem (2.4) as
and the second order energy of problem (2.4) as (if \(\mathcal {U}_{0}\in\mathcal{D(A)}\))
Lemma 3.4
Let \((\phi,\psi,\theta,z)\) be the solution of problem (2.4). Then the energy functional defined by (3.6) satisfies
with some constant \(C\geq0\).
Proof
Multiplying the first three equations in (2.4) by \(\gamma\phi_{t}\), \(\gamma\psi_{t}\), \(\beta\theta_{t}\), respectively, and integrating over \((0,1)\), and multiplying (2.4)4 by \(\frac{\xi z}{\tau}\) and integrating over \((0,1)\times(0,1)\) with respect to ρ and x, we get
For the last two terms on the right-hand side, by using Hölder’s inequality and Young’s inequality, we have
and
Combining (3.9)–(3.11), we obtain
The above assumption (3.2) implies that there exists a constant \(C\geq0\) such that
This gives (3.8). □
4 Energy decay result
In this section, we shall state and prove our decay result.
Theorem 4.1
Suppose that Assumption 2.1 holds and \(\mu_{1}>\mu_{2}\). For any initial value \(\mathcal{U}_{0}\in\mathcal{H}\), there exist positive constants C and α such that the energy of problem (2.4) satisfies
Moreover, if the initial value \(\mathcal{U}_{0}\in D(\mathcal{A})\), we have that, for some constants \(C>0\) and \(M_{1}>0\), the energy of problem (2.4) satisfies
In order to prove this result, we introduce various functionals and establish several lemmas. The construction of the auxiliary function \(I_{1}(t)-I_{3}(t)\), \(I_{5}(t)\) comes from [16].
Lemma 4.2
Let \((\phi,\psi,\theta,z)\) be the solution of (2.4). The functional \(I_{1}\) defined by
satisfies
Proof
By differentiating \(I_{1}\) and using (2.4), we conclude that
By using Young’s inequality and the fact \(\int_{0}^{1}\psi^{2}\,dx\leq\int _{0}^{1}\psi_{x}^{2}\,dx\), we have
For the fourth term in (4.2), using (2.2) and the generalized Hölder inequality, we obtain
By Sobolev–Poincaré inequality and \(\dot{E}(t)\leq0\), we get
in which \(C>0\) is a constant. Thus, together with the above two inequalities, Young’s inequality and the Sobolev embedding theorem for ψ, we obtain
Insert (4.3), (4.4), and (4.6) into (4.3), then Lemma 4.2 follows. □
Lemma 4.3
Let \((\phi,\psi,\theta,z)\) be the solution of (2.4). The functional \(I_{2}\) defined by
where g is the solution of
satisfies that, for any \(\varepsilon_{2} > 0\),
Proof
Using (2.4) and integration by parts, we conclude that
By the fact following from (4.8) that
we obtain that
By using Young’s inequality and (4.11)–(4.12), for any \(\varepsilon_{2} > 0 \), we have
Combining (4.13)–(4.16), we have (4.9). □
Lemma 4.4
Let \((\phi,\psi,\theta,z)\) be the solution of (2.4). The functional \(I_{3}\) defined by
satisfies for any \(\varepsilon_{3}>0\) that
Proof
By differentiating \(I_{3}\) and using (2.4), we conclude that
By using Young’s inequality and Poincaré’s inequality, we have
By using the fact that
we arrive at, for any \(\varepsilon_{3} > 0 \),
in which \(C_{2}\) is a positive constant. Combining (4.20)–(4.24) yields the conclusion. □
Next we deal with the boundary term in (4.18). We introduce the function
Lemma 4.5
Let \((\phi,\psi,\theta,z)\) be the solution of (2.4), then for \(\varepsilon_{3}>0\) the following estimate holds:
Proof
By using Young’s inequality, for \(\varepsilon_{3} > 0 \), we have
Also, we have
By using Young’s inequality and Poincaré’s inequality, for \(\varepsilon_{3} > 0 \), we have
Similarly,
Together with (4.27)–(4.29), using (4.23) gives us (4.26). □
Lemma 4.6
Let \((\phi,\psi,\theta,z)\) be the solution of (2.4). The functional \(I_{4}\) defined by
and its time derivative \(I'_{4}(t)\) satisfies
where \(C_{4}>0\) is the Sobolev embedding constant.
Proof
By differentiating \(I_{4}\) and using (2.4), we conclude that
Using Young’s inequality and Poincaré’s inequality clearly implies the conclusion (4.30). □
Lemma 4.7
Let \((\phi,\psi,\theta,z)\) be the solution of (2.4). The functional \(I_{5}\) defined by
for some constant \(m>0\) satisfies
Proof
By differentiating \(I_{5}\) and using (2.4), we conclude that
This gives (4.31). □
Now we define the Lyapunov functional \(\mathscr{L}(t)\) as follows:
where N, \(N_{2}\) are positive constants to be chosen properly later. For N large enough, it is not difficult to prove that there exist two positive constants \(\gamma_{1}\) and \(\gamma_{2}\) such that, for any \(t>0\),
Proof of Theorem 4.1
Combining Lemmas 4.2–4.7, we have
Firstly, we take \(\varepsilon_{3}\) small enough such that
Then we choose \(N_{2}\) so large that
thus we have
After that, we select \(\varepsilon_{2}\) small enough such that
and
Finally, we choose N so large that
Therefore (4.34) changes to
where M, \(M_{1}\) are positive constants.
Case 1: \({\frac{\rho_{1}}{K}=\frac{\rho_{2}}{b}}\).
In this case, (4.35) takes the form
Using (4.33), we get, for \(\alpha=\frac{M_{1}}{\gamma_{2}}\),
A simple integration of (4.36) over \((0,t)\) leads to
Recalling (4.33), we obtain
Case 2: \(0<|\frac{\rho_{1}}{K}-\frac{\rho_{2}}{b}|<\frac{M_{1}\gamma K}{4(K+b)}\).
Let
represent the first order energy defined in (3.6). By computation we have the estimate of the derivative of the second order energy (3.7) as
Let us estimate the last term in (4.35). By setting \(\Lambda =\frac{1}{K}(\frac{\rho_{2}K}{\rho_{1}}-b)\rho_{1}\neq0\) and using (2.4), (4.23), we have
Let
Then (4.35) becomes
for \(M_{2}=M_{1}-\frac{4|\Lambda|}{\gamma}(\frac{1}{K}+\frac{1}{b})>0\). Let
if \(N_{3}> \max\{C_{0}|\Lambda|-\gamma_{1},|\Lambda|,\frac{|\Lambda|}{4 C}\} \). Indeed, by using (4.11), (4.23), and \(ab\leq\frac {1}{2}a^{2}+\frac{1}{2}b^{2}\), we obtain
where \(C_{0}=\max\{\frac{3}{b\gamma},\frac{2}{K\gamma},\frac{1}{\rho _{2}\gamma}\}\). With the help of (4.33), we obtain
It is easy to prove that
for some positive constants \(c_{1}\) and \(c_{2}\). By using (4.39) and (4.40), we obtain
Thanks to the choice of \(N_{3}\), we have
Integrating (4.43) over \((0,t)\) yields
Using the fact that
we get that
This completes the proof of Theorem 4.1. □
References
Messaoudi, S.A., Said-Houari, B.: Energy decay in a Timoshenko-type system of thermoelasticity of type III. J. Math. Anal. Appl. 348, 298–307 (2008)
Messaoudi, S.A., Fareh, A.: Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds. Arab. J. Math. 2, 199–207 (2013)
Ayadi, M.A., Bchatnia, A., Hamouda, M., Messaoudi, S.: General decay in a Timoshenko-type system with thermoelasticity with second sound. Adv. Nonlinear Anal. 4, 263–284 (2015)
Grace, S.: Oscillation criteria for third order nonlinear delay differential equations with damping. Opusc. Math. 35, 485–497 (2015)
Öztürk, Ö., Akmon, E.: Nonoscillatory solutions of two dimensional nonlinear delay dynamical systems. Opusc. Math. 36, 651–669 (2016)
Racke, R.: Instability of coupled systems with delay. Commun. Pure Appl. Anal. 11, 1753–1773 (2012)
Nicaise, S., Pignotti, C., Valein, J.: Exponential stability of the wave equation with boundary time-varying delay. Discrete Contin. Dyn. Syst., Ser. S 4, 693–722 (2011)
Nicaise, S., Pignotti, C.: Stability of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21, 935–985 (2008)
Apalara, T.A., Messaoudi, S.A.: An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay. Appl. Math. Optim. 71, 449–472 (2015)
Guesmia, A., Messaoudi, S.A.: A general stability result in a Timoshenko system with infinite memory: a new approach. Math. Methods Appl. Sci. 37, 384–392 (2014)
Apalara, T.A., Messaoudi, S.A.: General stability result in a memory-type porous thermoelasticity system of type III. Arab J. Math. Sci. 20, 213–232 (2014)
Messaoudi, S.A., Said-Houari, B.: Uniform decay in a Timoshenko-type system with past history. J. Math. Anal. Appl. 360, 459–475 (2009)
Guesmia, A., Messaoudi, S.A.: On the stabilization of Timoshenko systems with memory and different speeds of wave propagation. Appl. Math. Comput. 219, 9424–9437 (2013)
Suh, H., Bien, Z.: Use of time delay action in the controller design. IEEE Trans. Autom. Control 25, 600–603 (1980)
Kafini, M., Messaoudi, S.A., Mustafa, M.I.: Well-posedness and stability results in a Timoshenko-type system of thermoelasticity of type III with delay. Z. Angew. Math. Phys. 66, 1499–1517 (2015)
Feng, B.W., Pelicer, M.L.: Global existence and exponential stability for a nonlinear Timoshenko system with delay. Bound. Value Probl. 2015, Article ID 206 (2015)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)
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The authors cordially thank the anonymous referees for their valuable comments and suggestions which led to the improvement of this paper.
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This work was partially supported by NNSF of China (61374089).
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Hao, J., Wei, J. Global existence and stability results for a nonlinear Timoshenko system of thermoelasticity of type III with delay. Bound Value Probl 2018, 65 (2018). https://doi.org/10.1186/s13661-018-0983-1
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DOI: https://doi.org/10.1186/s13661-018-0983-1