Global existence and stability results for a nonlinear Timoshenko system of thermoelasticity of type III with delay

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.

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.

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.

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). □

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. □

Acknowledgements

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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Funding

This work was partially supported by NNSF of China (61374089).

Affiliations

1. School of Mathematical Sciences, Shanxi University, Taiyuan, China

   • Jianghao Hao
   •  & Jing Wei

Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Jianghao Hao.

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Competing interests

The authors declare that they have no competing interests.

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