Abstract
This paper is concerned with the long-time behavior for a class of N-dimensional thermoelastic coupled structure equations with structural damping and past history thermal memory
This system arises from a model of the nonlinear thermoelastic coupled vibration structure with the clamped ends for simultaneously considering the medium damping, the viscous effect and the nonlinear constitutive relation and thermoelasticity based on a theory of non-Fourier heat flux laws. By considering the case where the internal (structural) damping is present, for \(0\leq\iota<1\), we show that the thermal source term \(f_{2}(\theta)\) is crucial to stabilizing the system and guarantees the existence of a global attractor for the above mentioned system in the present method.
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1 Introduction
In this paper, we study the N-dimensional nonlinear thermoelastic coupled structure equations with structural damping and past history thermal memory
which arise from a model of the nonlinear thermoelastic coupled vibration structure with the clamped ends for simultaneously considering the medium damping, the viscous effect and the nonlinear constitutive relation and thermoelasticity based on a theory of non-Fourier heat flux. The system is supplemented with the boundary conditions
for every \(t>0\), and the initial conditions
for every \(x\in\Omega\), where \(u^{0}(x)\), \(u^{1}(x)\) and \(\theta_{0}\) are assigned initial value functions.
Here the unknown variables \(u(x,t)\) and \(\theta(x,t)\) represent the vertical deflection of the structure and the vertical component of the temperature gradient, respectively. The subscript t denotes derivative with respect to t. Ω is a bounded domain of \(R^{N}\) with a smooth boundary ∂Ω, \(\sigma(\cdot)\) and \(\phi (\cdot)\) are the nonlinearity of the material and both continuous nonnegative nonlinear real functions, \(f_{1}(u)\) and \(f_{2}(\theta)\) are the source terms, \(k(s)\) is memory kernel and \(q_{1}(x)\) is the lateral load distribution, \(q_{2}(x)\) is the external heat supply, ν is a positive constant. What is more, the source terms \(f_{1}(u)\) and \(f_{2}(\theta)\) are essentially \(\vert u \vert ^{\rho}u\) and \(\vert \theta \vert ^{\varrho}\theta\), respectively, with \(0<\rho,\varrho\leq\frac{2}{N-2}\) if \(N\geq3\) and \(\rho, \varrho>0\) if \(N=1,2\), and the memory kernel \(k:R^{+}\mapsto R\) is assumed to be a positive bounded convex function vanishing at infinity and the assumptions on nonlinear functions \(\sigma(\cdot)\), \(\phi (\cdot)\), \(f_{1}(\cdot)\), \(f_{2}(\cdot)\) and the external force function \(q_{1}(x)\), \(q_{2}(x)\) will be specified later.
Without considering the thermal effect, this problem of the infinite dimensional dynamical systems determined by the elastic structure is based on the one-dimensional uncoupled beam equation
which was proposed as a model by introducing terms to account for effects of internal (structural) and external linear damping, and the stability theory under the clamped boundary conditions and the hinged boundary conditions was proved by Ball [1]. Ma and Narciso [2] proved the existence of global solutions and the existence of a global attractor for the Kirchhoff-type beam equation
with nonlinear external damping but without structural damping, subjected to the conditions
Without structural damping and thermal effects, this class of structure equations was studied by several authors, e.g., [3–7] and so on.
In the following we also make some comments about previous works for the long-time dynamics of thermoelastic coupled structure equations system with thermal effects.
As the case \(\iota=1\), Eq. (2) becomes the classical parabolic heat equation, thus thermoelastic coupled structure equations system without thermal memory term was considered by several authors. Giorgi et al. [8] studied a class of one-dimensional thermoelastic coupled beam equations with the classical parabolic heat equation but without structural damping
subjected to the hinged conditions
and gave the existence and uniqueness of global weak solution and the existence of global attractor. Berti et al. [9] studied a class of one-dimensional thermoelastic coupled beam equations with strong external damping and the classical parabolic heat equations
and proved the existence of solutions and the exponential decay property. In 2016, Fastovska [10] considered the existence of a compact global attractor for a nonlinear one-dimensional thermoelastic equation
with thermally insulated and clamped boundary conditions
and this system arose in phase transitions in rods made of shape memory alloys whose free energy density had a potential of Ginzburg-Landau form. In addition, Sprekels et al. [11] studied the dynamics of a nonlinear one-dimensional differential equation with the strain
arising from the study of phase transitions in shape memory alloys.
As the case \(0\leq\iota<1\), the memory term in Eq. (2) indicates the heat flux depending on the temperature gradient and its past history. When \(0\leq\iota<1\), Barbose et al. [12] studied the long-time behavior for a class of two-dimensional thermoelastic coupled plate equations
subjected to the hinged conditions
Potomkin [13] studied long-time behavior of thermoviscoelastic Berger two-dimensional plate equations
with boundary conditions
When the case \(\iota=0\), Wu [14] considered the following nonlinear plate equations with thermal memory effects due to non-Fourier heat flux laws:
subjected to the hinged conditions
and gave the existence of a global attractor.
In their works [12] and [13], they did not show directly that the system had a bounded absorbing set, because they found some technical difficulty due to the ‘extensibility’ term and θ. Instead, they showed that the system was gradient. While directly proving the existence of a bounded absorbing set in the present work, we mainly use Nakao’s lemma. The presence of \(f_{2}(\cdot)\) is crucial and guarantees the existence of a global attractor for the above mentioned system in the present method.
In addition, we also refer the reader to [15, 16] and the references therein for thermoelastic coupled structure equations.
It is well known that the infinite dimensional dynamical systems determined by the elastic structure are different because of the difference of boundary conditions. However, the above mentioned systems are all subjected to the hinged boundary conditions. For long-time dynamics for the thermoelastic coupled structure equations with clamped boundary, we refer the reader to [17].
In this paper, our fundamental assumptions on \(\sigma(\cdot)\), \(\phi(\cdot)\), \(f_{1}(\cdot)\), \(f_{2}(\cdot)\), \(g(\cdot)\), and \(q(x)\) are given as follows.
Assumption 1
We assume that \(\sigma(\cdot)\in C^{1}(R)\) satisfying
where \(\hat{\sigma}(z)=\int_{0}^{z}\sigma(s)\,ds\). This condition is promptly satisfied if \(\sigma(\cdot)\) is nondecreasing with \(\sigma(0)=0\).
Assumption 2
We also assume that \(\phi(\cdot)\in C^{1}(R)\) satisfying \(\phi(0)=0\) and \(\phi(\cdot)\) is nondecreasing and
Assumption 3
The function \(f_{1}(\cdot):R\longrightarrow R\) is of class \(C^{1}(R)\) and satisfies \(f(0)=0\), and there exist constants k and \(\rho>0\) such that
where \(\hat{f_{1}}(z)=\int_{0}^{z}f_{1}(s)\,ds\).
Assumption 4
The function \(f_{2}(\cdot ):R\longrightarrow R\) is of class \(C^{1}(R)\) and satisfies \(f_{2}(0)=0\), and there exist constants \(k_{2}\) and \(\varrho>0\) such that
Assumption 5
The assumptions on \(k(s)\) are as follows: k is vanishing at ∞; moreover,
where \(\mu\in C^{1}(R^{+})\cap L^{1}(R^{+})\), and there exists a constant \(\delta_{1}>0\), \(\forall s\in R^{+}\), such that
Assumption 6
\(q_{1}(x),q_{2}(x)\in L^{2}(\Omega)\).
Under the above assumptions, we prove the existence and uniqueness of global solutions and the existence of a global attractor for N-dimensional nonlinear thermoelastic coupled system (1)-(4) with structural damping and past history thermal memory.
2 Transformed system and basic spaces
Now, we observe that, because of the memory term with past history, problem (1)-(4) does not correspond to autonomous systems. Then we proceed as in Dafermos [18] and Giorgi [19] and Barbose et al. [12] and define a new variable \(\chi=\chi^{t}(x,s)\) by
From the definition of χ, for all \(t\geq0\), we have \(\chi^{t}(x,0)=0\), \(\Omega, t\in R^{+}\) and \(\chi^{0}(s)=\chi_{0}(s)\) in \(\Omega, s\in R^{+} \), where \(\chi_{0}(s)=\int_{0}^{s}\theta_{0}(\tau)\,d\tau\), \(s\in R^{+}\). Differentiate (19) with respect to t on both sides to get
Thus
and
where o is the history of θ.
Differentiate (19) with respect to s on both sides to get
Make the sum with (20) and (24) to get
So
Therefore thermal memory can be rewritten to be
Thus, from assumption (17) of kernel \(k(s)\), problem (1)-(4) is transformed into the new system
with the initial conditions
and homogeneous boundary conditions
Our analysis is based on the following Sobolev spaces. Let
and with respect to the new variable χ, we define the weighted space
which is a Hilbert space with the inner product and the norm defined by
Then, for regular solutions, we consider the phase space
In the case of weak solutions, we consider the phase space
In \(\mathbb{H}_{0}\) we adopt the norm defined by
3 The existence of global solutions and global attractor
Firstly, using the classical Galerkin method, we can establish the existence and uniqueness of regular solution and weak solution to problem (28)-(33) as in the work of Cavalcanti et al. [20] We state it as follows.
Theorem 7
Under Assumptions 1-6, for any initial data \((u^{0},u^{1},\theta^{0}, \chi^{0})\in\mathbb{H}_{1}\), problem (28)-(33) has a unique regular solution \((u,\theta,\chi)\) with
Theorem 8
Under the assumptions of Theorem 7, if the initial data \((u^{0},u^{1},\theta^{0},\chi^{0})\in\mathbb{H}_{0}\), there exists a unique weak solution \((u,\theta,\chi)\) of problem (28)-(33) such that
which depends continuously on the initial data with respect to the norm of \(\mathbb{H}_{0}\).
Remark 9
In both cases
where C is a constant and C denotes a different constant in different expression of this paper.
Remark 10
Theorem 8 implies that problem (28)-(33) defines a nonlinear \(C_{0}\)-semigroup \(S(t)\) on \(\mathbb{H}_{0}\). Indeed, let us set \(S(t)(u^{0},u^{1},\theta^{0},\chi_{0})=(u(t),u_{t}(t),\theta (t),\chi^{t})\), where u is the unique solution corresponding to the initial data \((u^{0},u^{1},\theta^{0},\chi_{0})\in\mathbb{H}_{0}\). Moreover, the operator \(S(t)\) defined in \(\mathbb{H}_{0}\) meets the usual semigroup properties \(S(t+\tau)=S(t)S(\tau)\), \(S(0)=I\), \(\forall{t,\tau\in{R}} \).
To prove the main result, we need the following Lemma 11 of Nakao and Lemma 12.
Lemma 11
[21]
Let \(\varphi(t)\) be a nonnegative continuous function defined on \([0,T)\), \(1< T\leq\infty\), which satisfies \(\sup_{t\leq s\leq t+1}\varphi(s)^{1+\eta}\leq M_{0}(\varphi (t)-\varphi(t+1))+M_{1}\), \(0\leq t\leq T-1\), where \(M_{0}\), \(M_{1}\), η are positive constants. Then we have
Lemma 12
[22]
Assume that for any bounded positive invariant set \(B\subset H\), and for any \(\varepsilon>0\), there exists \(T=T(\varepsilon,B)\) such that \(d(S(T)x,S(T)y)\leq\varepsilon+\varpi_{T}(x,y)\), \(\forall x,y\in B\), where \(\varpi_{T}:H\times H\rightarrow R\) satisfies, for any sequence \(\{ z_{n}\}\subset B\), \(\liminf_{m\rightarrow\infty}\liminf_{n\rightarrow\infty}\varpi _{T}(z_{n},z_{m})=0\). Then \(S(t)\) is asymptotically smooth.
The main result of an absorbing set reads as follows.
Theorem 13
Assume the hypotheses of Theorem 8, then the corresponding semigroup \(S(t)\) of problem (28)-(33) has an absorbing set \(\mathbb{B}\) in \(\mathbb{H}_{0}\).
Proof
Now we show that the semigroup \(S(t)\) has an absorbing set \(\mathbb{B}\) in \(\mathbb{H}_{0}\). Firstly, we can calculate the total energy functional
Let us fix an arbitrary bounded set \(B\subset\mathbb{H}_{0}\) and consider the solutions of problem (28)-(33) given by \((u(t),u_{t}(t),\theta(t),\chi)=S(t)(u^{0},u^{1},\theta^{0},\chi ^{0})\) with \((u^{0},u^{1},\theta^{0},\chi^{0})\in B\). Our analysis is based on the modified energy function
where \(\lambda_{1}>0\) is the first eigenvalue of the operator △ in \(H_{0}^{2}(\Omega)\). It is easy to see that \(\tilde{E}(t)\) dominates \(\Vert (u(t),u_{t}(t),\theta(t)) \Vert ^{2}_{\mathbb{H}_{0}}\) and \(\Vert \triangle u(s) \Vert ^{2}\leq4\tilde{E}(s)\).
By multiplying Eq. (28) by u and integrating over Ω
and inserting it into \(\tilde{E}(t)\), then integrating it from \(t_{1}\) to \(t_{2}\), and considering (12) and (15), we obtain that
where \(t_{1},t_{2}\in[t,t+1]\).
Now let us begin to estimate the right-hand side of (44) to use the above Lemma 11 of Nakao.
First, multiplying Eq. (28) by \(u_{t}\) and integrating over Ω, and multiplying Eq. (29) by θ and integrating over Ω, and taking the inner product with (30) by χ in \(L^{2}_{\mu}(R^{+},H_{0}^{1})\), then taking the sum and integrating from t to \(t+1\), we get
Using Schwarz’s inequality and Young’s inequality and Holder’s inequality (with \(\frac{\varrho}{\varrho+2}+\frac{2}{\varrho +2}=1\)), we get
With
as \(\varepsilon\leq\frac{k_{2}}{2}\), from (45) we have
Taking into account assumptions (13) and (16) of \(\phi(\cdot)\) and \(f_{2}(\cdot)\), we have
then we define an auxiliary function \(I^{2}(t)\) by putting
Since \(\frac{\iota}{2}\int_{t}^{t+1} \Vert \nabla\theta \Vert ^{2}\,d\tau\) is vanishing as \(\iota=0\), thus with \(0\leq\iota<1\), we can only get
So from the first inequality of (47), we have
Using twice Holder’s inequality with \(\frac{\varrho}{\varrho +2}+\frac{2}{\varrho+2}=1\), combined with assumption (16) on \(f_{2}(\theta)\) and the second inequality of (47), we have
In fact, as \(0<\iota<1\), we have \(\frac{\iota}{2}\int_{t}^{t+1} \Vert \nabla\theta \Vert ^{2}\,ds\leq I^{2}(t)\), thus we can directly get the estimate on \(\int_{t}^{t+1} \Vert \theta \Vert ^{2}\,ds\) without using assumption (16) on \(f_{2}(\theta)\).
Using the mean value theorem with \(\phi(0)=0\) and considering the estimate of (40), then using Young’s inequality combined with (47), we have
where \(\xi_{0}\) is among 0 and \(\int _{\Omega}\nabla u\nabla u_{t}\,dx\).
Considering assumption (18) on \(\mu(s)\), we have
Also, from the third inequality of (47), we have
Using Schwarz’s inequality and Young’s inequality, we get
Since (48), in view of the mean value theorem for integral, there exist number \(t_{1}\in[t,t+\frac{1}{4}]\) and number \(t_{2}\in[t+\frac{3}{4},t+1]\) such that
and
Thus from Schwarz’s inequality combined with (54)-(55), noting that \(\Vert \triangle u(s) \Vert ^{2}\leq4\tilde {E}(s)\), we have
Also, by Young’s inequality and considering (49), we have
Finally, using Young’s inequality again, we get that
Inserting (48)-(50), (52)-(53) and (56)-(58) into (44), we obtain
For the left-hand side of (59), we use the mean value theorem, then there exists number \(\tau\in[t_{1},t_{2}]\) such that
So we conclude that
Inserting (59) into (61), we obtain that
Letting \(0<\eta<\frac{1}{10}\), considering the boundedness of \(I(t)^{\frac{2\varrho}{\varrho+2}}\), then setting
from (62) we get
then (63) can be rewritten as
Using Nakao’s Lemma 11, we conclude that
As \(t\rightarrow\infty\), the first term on the right-hand side of (65) goes to zero, thus with \(\tilde{E}(t)\geq\frac {1}{4}( \Vert \triangle u \Vert ^{2}+ \Vert v \Vert ^{2}+ \Vert \theta \Vert ^{2}+ \Vert \chi \Vert ^{2}_{\mu})\), we conclude
is an absorbing set for \(S(t)\) in \(\mathbb{H}_{0}\). Theorem 13 is proved. □
The main result of a global attractor reads as follows.
Theorem 14
Assume the hypotheses of Theorem 8, then the corresponding semigroup \(S(t)\) of problem (28)-(33) is asymptotically compact in the space \(\mathbb{H}_{0}\).
Proof
We are going to apply Lemmas 11 and 12 to prove the asymptotic smoothness. Given the initial data \((u^{0},u^{1},\theta^{0},\chi ^{0})\) and \((v^{0},v^{1},\tilde{\theta}^{0},\tilde{\chi}^{0})\in B \) in a bounded invariant set \(B\subset\mathbb{H}_{0}\), let \((u,\theta,\chi )\), \((v,\tilde{\theta},\tilde{\chi})\) be the corresponding weak solutions of problem (28)-(33). Then the difference \(w=u-v\), \(\vartheta=\theta-\tilde{\theta}\), \(\Pi=\chi-\tilde{\chi}\) is the weak solutions of
where
Let us define
We can assume formally that w is sufficiently regular by density. Then, multiplying the first equation in (67) by \(w_{t}\) and integrating over Ω, and multiplying the second equation in (67) by ϑ and integrating over Ω, and taking the inner product with Π for the third equation in (67) in the space \(L^{2}_{\mu}(R^{+},H_{0}^{1})\), then taking the sum, we get
where
Let us estimate the right-hand side of (70).
Considering the continuity of \(\sigma'(\cdot)\) and estimate (40), we have
Applying the mean value theorem combined with estimate (40), by Young’s inequality we get
Also use the mean value theorem combined with estimate (40) and Young’s inequality to get
where \(\xi_{1}\) is among 0 and \(\int_{\Omega}\nabla u\nabla u_{t}\,dx\), and \(\xi_{2}\) is among \(\int_{\Omega}\nabla u\nabla u_{t}\,dx\) and \(\int _{\Omega}\nabla v\nabla v_{t}\,dx\).
By Holder’s inequality, Minkowski’s inequality combined with the estimate of (40), then by \(H_{0}^{1}(\Omega)\hookrightarrow \hookrightarrow L^{2(\rho+1)}(\Omega)\) (with \(0<\rho,\varrho\leq \frac{2}{N-2}\) if \(N\geq3\) and \(\rho, \varrho>0\) if \(N=1,2\)) and Young’s inequality, we obtain
On the other hand, combining with assumption (18) on \(\mu(s)\), we obtain
Considering assumption (17) on \(f_{2}(\cdot)\), we have
Thus, by inserting (72)-(77) into (70), we get that
Then, integrating from t to \(t+1\) and defining an auxiliary function \(F^{2}(t)\), we get
As \(\iota=0\), \(\iota\int_{t}^{t+1} \Vert \nabla\vartheta \Vert ^{2}\,d\tau\) is obsolescent, thus with \(0\leq\iota<1\) we can only get
Then, multiplying the first equation in (67) by w and integrating over Ω again, we obtain that
Integrating from \(t_{1}\) to \(t_{2}\), we get
Now let us estimate the right-hand side of (82). Firstly, from the first inequality of (80), we infer that
Thus there exist \(t_{1}\in[t,t+\frac{1}{4}]\) and \(t_{2}\in[t+\frac {3}{4},t+1]\) such that
then we can deduce that
Use Schwarz’s inequality and Young’s inequality to obtain
Apply the mean value theorem combined with estimate (40) to get
Assumption (14) on \(f_{1}(\cdot)\) and the estimate of (40) imply that
Using the mean value theorem and considering the assumption on \(\phi (\cdot)\) and the estimate of (40), we have
and
where \(\xi_{4}\) is among \(\int_{\Omega}\nabla u\nabla u_{t}\,dx\) and \(\int _{\Omega}\nabla v\nabla v_{t}\,dx\).
Finally, using twice Holder’s inequality with \(\frac{\varrho +1}{\varrho+2}+\frac{1}{\varrho+2}=1\), combined with the third inequality of (80), we have
Thus, by Schwarz’s inequality and Young’s inequality, we get
By inserting (83) and (85)-(90) and (92) into (82), we obtain that
Then from the definition of \(E_{w}(t)\) combined with (80), (83) and (91), we obtain that
For (94), by using the mean value theorem, there exists \(t^{\ast}\in[t_{1},t_{2}]\) such that
From (79), we see that
Let \(E_{w}(\tau)=\sup_{t\leq\sigma\leq t+1} E_{w}(\sigma)\) with \(\tau\in[t,t+1]\), then integrate (78) over \([t,\tau]\) and over \([t^{*},t+1]\) to have
Inserting (95) into (97), we obtain
Therefore, from the boundary of \(F(t)^{\frac{2\varrho}{\varrho+2}}\), we have
Therefore
From Nakao’s Lemma 11, there exist \(C_{B}>0\) and \(C_{T}>0\) such that
From the definition of \(E_{w}(t)\), we have
Given \(\varepsilon>0\), we choose T large such that
and define \(\varpi_{T}:\mathbb{H}_{0}\times\mathbb{H}_{0}\rightarrow R\) as
for all \((u^{0},u^{1},\theta^{0},\chi^{0}),(v^{0},v^{1},\tilde{\theta}^{0},\tilde {\chi}^{0})\in B\).
Let \((u_{n}^{0},u_{n}^{1},\theta^{0}_{n},\chi^{0}_{n})\) be a given sequence of initial data in B. Then the corresponding sequence \((u_{n},u_{tn},\theta_{n},\chi_{n})\) of solutions of problem (28)-(33) is uniformly bounded in \(\mathbb{H}_{0}\), because B is bounded and positively invariant. So \(\{u_{n}\}\) is bounded in \(C([0,\infty ),H_{0}^{2}(\Omega))\cap C^{1}([0,\infty),L^{2}(\Omega))\). Since \(H_{0}^{2}(\Omega)\hookrightarrow\hookrightarrow H_{0}^{1}(\Omega)\) compactly, there exists a subsequence \({u_{nk}}\) which converges strongly in \(C([0,T],H_{0}^{1}(\Omega))\). Therefore
and
So \(S(t)\) is asymptotically smooth in \(\mathbb{H}_{0}\). That is, Lemma 12 holds. Thus Theorem 14 is proved. □
Theorem 15
The corresponding semigroup \(S(t)\) of problem (28)-(33) has a compact global attractor in the phase space \(\mathbb{H}_{0}\).
Proof
In view of Theorems 8, 13 and 14, we directly get Theorem 15. □
4 Conclusion
In this paper, we gave the existence and uniqueness of global solutions and the existence of a global attractor in \(\mathbb{H}_{0}\) for an N-dimensional nonlinear thermoelastic coupled system with structural damping and past history thermal memory
By considering the case where the internal (structural) damping is present, for \(0\leq\iota<1\), we show that the thermal source term \(f_{2}(\theta)\) is crucial and guarantees the existence of a global attractor for the above mentioned system in the present method. This main result may provide the design basis for the thermoelastic coupled structure in engineering. The relevant results have been mentioned in Introduction of [12] and [13].
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The project is supported by the National Natural Science Foundation of China (Grant No.11172194, the role of the funding lies in the collection of data and the analysis of the paper), and the Natural Science Foundation of Shanxi Province, China (Grant No. 2015011006, the role of the funding lies in writing the manuscript).
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This paper does not involve conflict of interests between the authors, and all authors declare that they have no competing interests.
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This paper is mainly completed by DX. JW deals with the structural damping term as proving the existence of a bounded absorbing set. All authors read and approved the final manuscript.
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Wang, D., Zhang, J. Long-time dynamics of N-dimensional structure equations with thermal memory. Bound Value Probl 2017, 136 (2017). https://doi.org/10.1186/s13661-017-0864-z
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DOI: https://doi.org/10.1186/s13661-017-0864-z