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Global existence of solutions for a class of thermoelastic plate systems
Boundary Value Problems volume 2018, Article number: 161 (2018)
Abstract
This paper is concerned with the initial-boundary value problem for a class of thermoelastic plate systems. Under some appropriate assumptions, the global existence of solutions is obtained.
1 Introduction
In this paper, we study the following initial-boundary value problem for a class of thermoelastic plate systems:
with boundary conditions
and initial conditions
where Ω is a bounded domain of \({\mathbb {R}}^{N}\) (\(N\geq1\)) with a smooth boundary ∂Ω, \(0\leq\omega<1\), and \(\nu>0\). The function g, external force f and memory kernel k will be specified later.
It is well known that temperature gradients in a plate will contribute to plate deformation. Problem (1.1)–(1.3) can be used to describe the deformation and the temperature distribution of a homogeneous, isotropic and thermoelastic thin material with memory, see [1, 2] for the details. Functions \(u(x,t)\) and \(\theta(x,t)\) represent the displacement and temperature variation field relative to the equilibrium reference value, respectively. The cases \(\omega=0\) and \(0<\omega<1\) in (1.1)2 are usually referred to as the Gurtin–Pipkin model [3] and the Coleman–Gurtin model [4], respectively. In the absence of thermal effects, (1.1) reduces to an extensible plate equation. This class of equations model the vibrations of extensible elastic beams (when \(N=1\)) and plates (when \(N=2\)), and have been extensively investigated (see, e.g., [5–11] and the references therein).
Wu [12] studied
subject to (1.2) and (1.3). Under the assumption \(f\in C^{2}\), the author obtained the global existence and uniqueness of solutions, as well as the existence of global attractors. Moreover, when f is assumed real analytic, the convergence of global solutions to a single steady state, as time goes to infinity, was proved and also an estimate of the convergence rate was provided. Barbosa and Ma [13] investigated problem (1.1)–(1.3) by adding an extra external force h to (1.1)1. Using the assumptions \(g,f\in C^{1}\), the authors derived the global well-posedness of solutions, the existence of global attractors with finite fractal dimension, and the existence of exponential attractors.
In the present paper, our purpose is to tackle the global existence of solutions to problem (1.1)–(1.3) under weaker assumptions on g and f. As in [12, 13], we employ the past history approach [14, 15], so that problem (1.1)–(1.3) can be transformed into an equivalent system in the history phase space. By means of the potential well theory [16, 17], we establish the theorems on global existence of solutions by discussing the level of initial energy.
This paper is organized as follows. In Sect. 2, some assumptions on g, f and k are displayed. Moreover, problem (1.1)–(1.3) is transformed into an equivalent system, and the main results of this paper are stated. In Sect. 3, the global existence of solutions with subcritical initial energy is established. In Sect. 4, the global existence of solutions with critical initial energy is derived.
2 Preliminaries and main results
2.1 Notations and assumptions
Throughout the paper, for simplicity, we denote
Moreover, \((\cdot,\cdot)\) denotes either the \(L^{2}\)-inner product or a duality pairing between a space and its dual space.
We make the following assumptions on g, f and k, respectively.
- \((\mathrm{A}_{1})\) :
-
\(g\in C(\mathbb{R})\), \(g(z)>0\), and there exists a constant \(\alpha>0\) such that \(\alpha G(z)\geq zg(z)\), where
$$G(z)= \int_{0}^{z} g(s)\, \mathrm{d}s. $$ - \((\mathrm{A}_{2})\) :
-
\(f\in C(\mathbb{R})\). There exists a constant \(\beta>0\) such that \(|f(u)|\leq\beta|u|^{p-1}\), where
$$2\leq p< \infty \quad \mbox{if } N\leq4,\qquad 2\leq p< \frac{2N}{N-4}\quad \mbox{if } N>4. $$Moreover, there exists a constant \(\gamma>2\tilde{\alpha}\) such that \(uf(u)\geq\gamma F(u)\), where \(\tilde{\alpha}:=\max\{1,\alpha\}\) and
$$F(u)= \int_{0}^{u} f(s)\, \mathrm{d}s. $$ - \((\mathrm{A}_{3})\) :
-
\(k\in C^{2}(\mathbb{R}^{+})\), \(k(\tau)\geq0\), \(k^{\prime}(\tau )\leq0\) and \(k^{\prime\prime}(\tau)\geq0\) for all \(\tau\in\mathbb {R}^{+}\). In addition, \(\mu(\tau):=-(1-\omega)k^{\prime}(\tau)\).
2.2 Reformulation of the problem
We define the auxiliary variable
Thus
Consequently, in view of [15, p. 165], problem (1.1)–(1.3) is transformed into the following equivalent system:
with boundary conditions
and initial conditions
where
2.3 Statement of main results
We introduce a weighted \(L^{2}\)-space
which is a Hilbert space equipped with the inner product
and the norm
Definition 2.1
\((u(t),\theta(t),v^{t})\) is called a weak solution to problem (2.1)–(2.3) if \(u\in L^{\infty}(0,T;H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega))\), \(u_{t}\in L^{\infty}(0,T;H_{0}^{1}(\varOmega))\), \(\theta\in L^{\infty}(0,T;L^{2}(\varOmega))\), \(\psi^{t}\in L^{\infty}(0,T;\mathcal{L})\), \(u(x,0)=u_{0}(x)\), \(u_{t}(x,0)=u_{1}(x)\), \(\theta(x,0)=\theta_{0}(x)\), \(\psi ^{0}(x,\tau)=\psi_{0}(x,\tau)\), and
for any \(\varphi_{1}\in H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega)\), \(\varphi_{2}\in H_{0}^{1}(\varOmega)\), \(\varphi_{3}\in\mathcal{L}\) and a.e. \(t\in(0,T]\).
The energy associated with problem (2.1)–(2.3) is given by
Furthermore, we define the energy functional
and the Nehari functional
Thus, all nontrivial stationary solutions belong to the Nehari manifold defined by
where \(\mathcal{H}:=H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega)\times L^{2}(\varOmega )\times\mathcal{L}\). Then the mountain pass level of J can be characterized as
We introduce the potential well
and let
The main results of this paper are stated as follows.
Theorem 2.1
Let assumptions \((\mathrm{A}_{1})\)–\((\mathrm{A}_{3})\) be fulfilled, \(u_{0}\in H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega)\), \(u_{1}\in H_{0}^{1}(\varOmega)\), \(\theta_{0}\in L^{2}(\varOmega)\), \(\psi_{0}\in\mathcal{L}\). Assume that \(0< E(0)< d\), and \(I(u_{0},\theta_{0},\psi_{0})>0\) or \((u_{0},\theta_{0},\psi_{0})=(0,0,0)\). Then problem (2.1)–(2.3) admits a global solution \((u,\theta ,\psi^{t})\in\mathcal{W}\). Moreover,
Theorem 2.2
Let assumptions \((\mathrm{A}_{1})\)–\((\mathrm{A}_{3})\) be fulfilled, \(u_{0}\in H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega)\), \(u_{1}\in H_{0}^{1}(\varOmega)\), \(\theta_{0}\in L^{2}(\varOmega)\), \(\psi_{0}\in\mathcal{L}\). Assume that \(E(0)=d\) and \(I(u_{0},\theta_{0},\psi_{0})\geq0\). Then problem (2.1)–(2.3) admits a global solution \((u,\theta,\psi^{t})\in\overline{\mathcal{W}}\).
3 Proof of Theorem 2.1
Let \(\{w_{j}\}_{j=1}^{\infty}\) be an orthogonal basis of \(H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega)\) and let an orthonormal basis of \(L^{2}(\varOmega)\) be given by eigenfunctions of
Then \(\{v_{j}\}_{j=1}^{\infty}\) is an orthonormal basis of \(H_{0}^{1}(\varOmega)\), where \(v_{j}=\frac{w_{j}}{\lambda_{j}^{\frac{1}{4}}}\). We select \(\{e_{j}\}_{j=1}^{\infty}\) as \(\{l_{k}v_{j}\}_{k,j=1}^{\infty}\), where \(\{l_{k}\} _{k=1}^{\infty}\) is an orthonormal basis of \(L_{\mu}^{2}(\mathbb{R}^{+})\). Then \(\{e_{j}\}_{j=1}^{\infty}\) is an orthonormal basis of \(\mathcal{L}\).
We construct the approximate solutions to problem (2.1)–(2.3) as
which satisfy
with
The approximate problem (3.1)–(3.2) can be reduced to an ordinary differential system in the variables \(\xi_{jn}(t)\), \(\eta _{jn}(t)\) and \(\zeta_{jn}(t)\). In terms of standard theory for ODEs, there exists a solution \((u_{n}(t),\theta_{n}(t),\psi^{t}_{n})\) on some interval \([0,T_{n})\) with \(T_{n}\leq T\). The following estimates will allow us to extend the local solutions to \([0,T]\) for all \(T>0\).
Multiplying (3.1)1 by \(\xi_{jn}^{\prime}(t)\), (3.1)2 by \(\eta_{jn}(t)\), and (3.1)3 by \(\zeta_{jn}(t)\), summing over j, and adding the two results, we obtain
where
Since \(\psi^{t}_{n}(x,0)=0\), we deduce from \((\mathrm{A}_{3})\) that
Hence, by integrating (3.3) with respect to t from 0 to t, we get
We now claim that
for all \(t\in[0,T]\) and sufficiently large n.
Indeed, if \((u_{0},\theta_{0},\psi_{0})=(0,0,0)\), then \((u_{0},\theta_{0},\psi _{0})\in\mathcal{W}\). If \(I(u_{0},\theta_{0},\psi_{0})>0\), then, from \(E(0)< d\), i.e.,
it follows that \(J(u_{0},\theta_{0},\psi_{0})< d \). Hence \((u_{0},\theta_{0},\psi _{0})\in\mathcal{W}\). Thus \((u_{n}(0),\theta_{n}(0),\psi^{0}_{n})\in\mathcal {W}\) for sufficiently large n due to (3.2). As a result, assertion (3.6) follows as desired. If it was not the case, there would exist a \(0< t_{0}< T\) such that \((u_{n}(t_{0}),\theta_{n}(t_{0}),\psi ^{t_{0}}_{n})\in\partial\mathcal{W}\), i.e., \(I(u_{n}(t_{0}),\theta_{n}(t_{0}),\psi ^{t_{0}}_{n})=0\) and \((u_{n}(t_{0}),\theta_{n}(t_{0}),\psi^{t_{0}}_{n})\neq(0,0,0)\), or \(J(u_{n}(t_{0}),\theta_{n}(t_{0}),\psi^{t_{0}}_{n})=d\). Due to (3.4), (3.5) and (3.2), we get
for all \(t\in[0,T]\) and sufficiently large n. This tells us that \(J(u_{n}(t_{0}),\theta_{n}(t_{0}),\psi^{t_{0}}_{n})=d\) is impossible. On the other hand, if \(I(u_{n}(t_{0}),\theta_{n}(t_{0}),\psi^{t_{0}}_{n})=0\) and \((u_{n}(t_{0}),\theta_{n}(t_{0}),\psi^{t_{0}}_{n})\neq(0,0,0)\), then, by the definition of d, we get \(J(u_{n}(t_{0}),\theta_{n}(t_{0}),\psi^{t_{0}}_{n})\geq d\), which contradicts (3.7).
We deduce from \((\mathrm{A}_{1})\) and \((\mathrm{A}_{2})\) that
Combining this with (3.4)–(3.6) and (3.2), we arrive at
for all \(t\in[0,T]\) and sufficiently large n. Moreover,
where \(q=\frac{p}{p-1}\), and \(C_{\ast}\) is the constant for the Sobolev embedding \(H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega)\hookrightarrow L^{p}(\varOmega)\).
Hence there exist \((u,\theta,\psi^{t})\) and subsequences of \(\{u_{n}\}\), \(\{ \theta_{n}\}\), \(\{\psi^{t}_{n}\}\), still represented by the same notations (and we shall not repeat this), such that, as \(n\to\infty\),
for any \(T>0\). In view of [18, Lemma 1.3], we have \(\chi=f(u)\). According to the Aubin–Lions lemma, we have
which, together with (3.8), (3.9) and (3.11), gives
Therefore, by the arguments similar to the proof given in [14, p. 343–345], we can pass to the limit in the approximate problem (3.1)–(3.2). Thus \((u,\theta,\psi^{t})\in\mathcal {W}\) is a global solution to problem (2.1)–(2.3).
Next, we prove (2.4). Indeed, note that
where \(\vartheta_{1}=\sigma\|\nabla u_{n}\|^{2}+(1-\sigma)\|\nabla u\|^{2}\), \(0<\sigma<1\). Hence it follows from (3.14) that
Furthermore,
where \(\vartheta_{2}=\sigma u_{n}+(1-\sigma)u\). This, together with (3.10), yields
Consequently, by (3.9), (3.11)–(3.13), (3.5), (3.15), (3.16) and (3.2), we obtain
Thus the proof of Theorem 2.1 is complete.
4 Proof of Theorem 2.2
We divide the proof of this theorem into two cases.
Case 1. \((u_{0},\theta_{0},\psi_{0})\neq(0,0,0)\).
Let \(\delta_{m}=1-\frac{1}{m}\), \(u_{m0}=\delta_{m} u_{0}\), \(\theta_{m0}=\delta_{m} \theta_{0}\) and \(\psi_{m0}=\delta_{m} \psi_{0}\), \(m=2,3,\dots\) . We consider problem (2.1)–(2.2) with the following initial conditions:
From \(I(u_{0},\theta_{0},\psi_{0})\geq0\),
and
it is easy to verify that there exists a unique \(\delta_{\ast}=\delta _{*}(u_{0})\geq1\) such that \(J(\delta u,\delta\theta,\delta\psi^{t})\) is strictly increasing for \(\delta\in[0,\delta_{\ast}]\) and assumes the maximum at \(\delta=\delta _{\ast}\). Hence \(J(u_{m0},\theta_{m0},\psi_{m0})< J(u_{0},\theta_{0},\psi_{0})\) and \(I(u_{m0},\theta_{m0},\psi_{m0})> 0\). Moreover,
We further obtain
and
Hence, we conclude from Theorem 2.1 that problem (2.1)–(2.2) and (4.1) admits a global solution \((u_{m}(t),\theta_{m}(t),\psi^{t}_{m})\in\mathcal{W}\) satisfying
with
and
Consequently,
By the arguments similar to the proof of Theorem 2.1, we see that problem (2.1)–(2.3) admits a global solution \((u,\theta,\psi^{t})\in\overline{\mathcal{W}}\).
Case 2. \((u_{0},\theta_{0},\psi_{0})=(0,0,0)\).
In this case, it is clear that \(J(u_{0},\theta_{0},\psi_{0})=0\). Thus
Let \(\delta_{m}=1-\frac{1}{m}\) and \(u_{m1}=\delta_{m} u_{1}(x)\), \(m>1\) and consider problem (2.1)–(2.2) with the following initial conditions:
Note that
We conclude from Theorem 2.1 that problem (2.1)–(2.2) and (4.2) admits a global solution \((u_{m},\theta_{m},\psi ^{t}_{m})\in\mathcal{W}\). The remainder of the proof is the same as in Case 1.
Abbreviations
- \(\|u\|_{p}\) :
-
\(\|u\|_{L^{p}(\varOmega)}\)
- \(\|u\|\) :
-
\(\|u\|_{2}\)
- \(\mathcal{L}\) :
-
\(L_{\mu}^{2}(\mathbb{R}^{+};H_{0}^{1}(\varOmega))\)
- \(\mathcal{H}\) :
-
\(H^{2}(\varOmega)\cap H_{0}^{1}(\varOmega)\times L^{2}(\varOmega )\times\mathcal{L}\)
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The author would like to thank the reviewers for the valuable suggestions.
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This work is supported by the Science and Technology Plan Project of Gansu Province in China (Grant No. 17JR5RA279).
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Liu, Y. Global existence of solutions for a class of thermoelastic plate systems. Bound Value Probl 2018, 161 (2018). https://doi.org/10.1186/s13661-018-1081-0
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DOI: https://doi.org/10.1186/s13661-018-1081-0