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Global attractors for Kirchhoff wave equation with nonlinear damping and memory
Boundary Value Problems volume 2020, Article number: 116 (2020)
Abstract
In this paper, we prove that the existence of global attractors for a Kirchhoff wave equation with nonlinear damping and memory.
1 Introduction
Let Ω be an open bounded subset of \(\mathbb{R}^{N}\) with sufficiently smooth boundary Γ, we consider the following Kirchhoff wave equation with nonlinear damping and linear memory:
where \(M(s)=1+s^{\frac{m}{2}} \), \(m\geq 1\), \(k(0)\), \(k(\infty )>0\) and \(k'(s)\leq 0\) for every \(s\in \mathbb{R}^{+}\), and the assumptions on nonlinear functions \(f(u)\), \(g(u_{t})\), \(a(x)\) and external force term \(h(x)\) will be specified later.
This kind of wave models goes back to Kirchhoff. In 1883, Kirchhoff [1] firstly introduced the following equation to describe small vibrations of an elastic stretch string:
where \(M(s)=a+bs\). There has been much research on global attractors; Lazo studied the existence for the IBVP of the Kirchhoff equation with memory term [2]
Chueshov [3] studied the well-posedness and the global attractors of the Kirchhoff equation with strong nonlinear damping
Next, Chueshov [4] also studied the Kirchhoff equation with strong nonlinear damping in nature space \(\mathcal{H}=H_{0}^{1}(\varOmega )\cap L^{p+1}(\varOmega )\times L^{2}( \varOmega )\) as \(\theta =1\). For related work on the Kirchhoff wave equations with strong damping, see [5, 6] and the references therein.
When \(M(s)=0\), Eq. (1) become the well-known wave equation. Ma and Zhong [7] showed the existence of global attractors for the hyperbolic equation with memory
Recently, Park and Kang [8] studied the existence of global attractors for the semilinear hyperbolic with nonlinear damping and memory
In [9], Kang and Rivera showed the existence of global attractors for the beam equation localized nonlinear damping and memory
Motivated by [5, 7–9], we will prove the existence of global attractors for Eq. (1).
Following the framework proposed in [7], we shall add a new variable η to the system, which corresponds to the relative displacement history. Let us define
By differentiation, we have
Let \(\mu (s)=-k'(s)\), \(k(\infty )=1\), (1) transforms into the following system:
with boundary condition
and initial conditions
This paper is organized as follows. In Sect. 2, we introduce some preliminaries. In Sect. 3, we show the existence of a bounded absorbing set in \(\mathcal{H}\). In Sect. 4, we give the existence of global attractors of problems (6)–(9).
2 Preliminaries
We first state some assumptions, which will be used in this paper.
Assumption (1)
The memory kernel μ is required to satisfy the following hypotheses:
- (h1):
\(\mu (s)\in C^{1}(\mathbb{R})\cap L^{1}(\mathbb{R})\), \(\forall s\in \mathbb{R}^{+}\);
- (h2):
\(\int ^{\infty }_{0}\mu (s)\,ds=k(0)\);
- (h3):
\(\mu (s)\geq 0\), \(\mu '(s)\leq 0\);
- (h4):
\(\mu '(s)+k_{1}\mu (s)\leq 0\), \(\forall s\in \mathbb{R}^{+}\), for some \(k_{1}>0\).
Assumption (2)
The function \(a(x)\) satisfies
where \(\alpha _{0}\) is a constant.
Assumption (3)
The function \(f\in C^{1}(\mathbb{R})\) satisfies
where \(0< p<\infty \), if \(n\leq 2\), and \(0< p\leq \frac{2}{n-2}\) if \(n\leq 2\). \(\lambda _{1}\) is the constant in the Poincáre type inequality \(\|\nabla u\|^{2}\geq \lambda _{1}\|u\|^{2}\).
Assumption (4)
The damping function \(g\in C^{1}(\mathbb{R})\) satisfies
with \(1\leq q<\infty \) if \(n\leq 2\), and \(1\leq q\leq \frac{n+2}{n-2}\) if \(n>2\).
In order to consider the relative displacement η as a new variable, we introduce the weighted \(L^{2}\)-space
which is a Hilbert space endowed with inner product and norm
respectively.
Our analysis is given on the phase space
which is equipped with the norm
In order to obtain the global attractors of the problems (6)–(9), we need the following theorem of existence, uniqueness of solution and continuous dependence on the initial data.
Theorem 2.1
([9])
Let assumptions(1)–(4)hold, if\(z_{0}=(u_{0},v_{0},\eta _{0})\in \mathcal{H}\), then there exists a unique solution\(z=(u,u_{t},\eta )\)of (6)–(9) such that
Next,we recall the simple compactness criterion stated in [9, 10].
Definition 2.1
Let X be a Banach space and B be a bounded subset of X, we call a function \(\varPhi (\cdot ,\cdot )\) which defined on \(X\times X\), is a contractive on \(B\times B\) if for any sequence \(\{x_{n}\}^{\infty }_{n=1}\subset B\), there is a subsequence \(\{x_{n_{k}}\}^{\infty }_{k=1}\subset \{x_{n}\}^{\infty }_{n=1}\) such that
Denote all such contractive functions on \(B\times B\) by \(C(B)\).
Theorem 2.2
Let\(\{s(t)\}_{t\geq 0}\)be a semigroup on a Banach space\((X,\| \cdot \|)\)and has a bounded absorbing set\(B_{0}\). Moreover, assume that for any\(\varepsilon \geq 0\)there exist\(T=T(B_{0},\varepsilon )\)and\(\varPhi (\cdot ,\cdot )\in C(B)\)such that
where\(\varPhi _{T}\)depends onT. Then\(\{s(t)\}_{t\geq 0}\)is asymptotically compact inX, i.e., for any bounded sequence\(\{y_{n}\}_{n}^{\infty }\subset X\)and\(\{t_{n}\}\)with\(t_{n}\rightarrow \infty \), \(\{S(t_{n})y_{n}\}_{n=1}^{\infty }\)is compact inX.
Lemma 2.1
([11])
Let\(g(\cdot )\)satisfy condition (13). Then for any\(\delta > 0\)there exists\(c(\delta )>0\), such that
3 Absorbing set in \(\mathcal{H}\)
In this section, we prove the existence of the bounded absorbing set in \(\mathcal{H}\). We use \(C_{i}\) to denote several positive constants.
Lemma 3.1
Under assumptions(1)–(4), the semigroup\(\{S(t)\}_{t\geq 0}\)corresponding to problems (6)–(9) has a bounded absorbing set in\(\mathcal{H}\).
Proof
we take the scalar product in \(L^{2}\) of system (6) with \(u_{t}\) and (7) with η, respectively, we have
where \(F(u)=\int _{0}^{u}f(s)\,ds\). As in [7]
We set
Then from (17) and (18) we obtain
By the hypothesis (12) we know that there are \(\lambda >\lambda _{1}> 0\) and \(C_{0}\) such that
Using the Young inequality, we have
we choose proper λ and ε small enough so that \(\frac{1}{2}-\frac{\lambda }{4\lambda _{1}}-\varepsilon >\frac{1}{8}\), and we have
combining (19) with (22), we have
Taking the scalar product in \(L^{2}\) of (6) with \(v=u_{t}+\varepsilon u\), we obtain
Let
so
Similarly, using (21), the Poincáre inequality and the Young inequality, choosing proper λ and ε small enough so that \(\frac{1}{2}-\frac{\varepsilon ^{2}}{2\lambda _{1}}- \frac{\lambda }{4\lambda _{1}}-\varepsilon >\frac{1}{8}\), we have
It is obvious that (10) and (13) imply that there are \(\varepsilon >0\) and \(C>0\) such that
Due to the Young inequality we have
so
where C is a constant which is independent of s.
Then from (29), using the Hölder inequality, the Young inequality and the Sobolev embedding \(H^{1}_{0}(\varOmega )\hookrightarrow L^{q+1}(\varOmega )\), we obtain
where \(a_{0}=\sup_{x\in \varOmega }{a(x)}\), and γ is a constant. From (21), (27), (28), (30) we have
we choose ε and C small enough so that \(\frac{1}{2}-\frac{k(0)\varepsilon ^{2}}{k_{1}}-C>\frac{1}{4}\), we get
where \(C_{E(0)}\) is a constant which depends on ε, γ, C and \(E(0)\), \(C'_{\varepsilon }\) is a constant depending on ε, \(C_{\delta }\) and C.
We have
where \(C_{0}=\max \{2,1+\frac{2\delta ^{2}}{\lambda _{1}}\}\).
Integrating (25), combining with (23), (26), (31), yields
where \(\delta '=\min \{\delta ,k_{1}\}\). Therefore, for any \(\rho >\frac{4C'_{\varepsilon }(\operatorname{mes}(\varOmega )+\|h\|^{2})}{\delta '}\) there exists \(t_{0} \) such that
Set
then we see \(B_{0}\) is a bounded absorbing set. Define
so \(B_{1}\) is also a bounded absorbing set. □
4 Existence of the global attractor in \(\mathcal{H}\)
4.1 A priori estimate
Firstly, we use the prior estimates to obtain the asymptotic compactness following the standard energy method. In this section, \(C_{i}\) are positive constants.
Let \((u,u_{t},\eta )\) and \((v,v_{t},\xi )\) be two solution to systems (6)–(9), and \((u,u_{t},\eta )\) and \((v,v_{t},\xi )\in B_{1}\), \(\omega (t)=u(t)-v(t)\), \(\zeta =\eta -\xi \). Then \(\omega (t)\), ζ satisfy
firstly, taking the scalar product in \(L^{2}\) of (35) with ω and integrating over \([0,T]\), we get
Using the Young inequality and \((h3)\), we obtain
Secondly, taking the scalar product in \(L^{2}\) of (35), (36) with \(\omega _{t}\) and integrating over \([0,T]\), we get
Let
Integrating (39) over \((s,T]\) and combining with (38), where \(s\in [0,T]\), we have
Integrating (40) over \([0,T]\) with respect to s, we get
Due to (10), (40), and Lemma 2.1, we obtain, for any \(\delta >0\),
where \(C_{2}\) is a constant which depends on δ, \(\alpha _{0}\) and \(k_{1}\).
Thus, from (37), (38) and (42) we have
where \(C_{3}=\max \{\frac{3}{2},\frac{k(0)+1}{2}\}\). From (23) and the existence of the absorbing set, we get
where \(C_{\rho }\) is a constant which depends on \(\operatorname{mes}(\varOmega )\), \(\|h\|^{2}\) and the size of \(B_{0}\). By a similar method to that of (30) and (43), (44), we have
similarly
combining (41), (43), (46), (47), we have
where
Then we have
4.2 Asymptotic compactness
In this subsection, following the argument in [9, 10], we will prove the asymptotic compactness of the semigroup \(\{S(t)\}_{t\geq 0}\) in \(\mathcal{H}\), which is given in the following theorem.
Theorem 4.1
Under assumptions(1)–(4), the semigroup\(\{S(t)\}_{t\geq 0}\)to systems (6)–(9) is asymptotically compact in\(\mathcal{H}\).
Proof
since the semigroup \(\{S(t)\}_{t\geq 0}\) has a bounded absorbing set, for every fixed \(\varepsilon >0\), we can choose that \(\delta \leq \frac{\varepsilon }{2C_{3}\operatorname{mes}(\varOmega )}\), and then let T become so large that
Hence, thanks to Theorem 2.2, we only need to verify that the function \(\varPhi _{T}(z_{0}^{1},z_{0}^{2})\) defined in (50) belongs to \(C(B_{1})\) for each fixed T. and we claim that
Here \((u(t),u_{t}(t),\eta )=S(t)z_{0}^{1}\) and \((v(t),v_{t}(t),\xi )=S(t)z_{0}^{1}\) are the solutions of (6)–(9) with respect to initial \(z_{0}^{1},z_{0}^{2}\in B_{1}\). Then, since \(C(B_{1})\) is a bounded positively invariant set in \(\mathcal{H}\), it follows that \((u_{n},u_{n_{t}},\eta ^{n})\) is uniformly bounded in \(\mathcal{H}\). We have
Then, by the compact embedding \(H_{0}^{1}(\varOmega )\hookrightarrow L^{k}(\varOmega )\), we have
where \(k\leq \frac{2n}{n-2}\), therefore from (56) we have
then from (57) and (10), we obtain
Finally, we follow a similar argument to the ones given in [9, 10]. We have
Finally, combining (58)–(65) we get \(\varPhi (\cdot ,\cdot )\in C(B_{1})\). □
4.3 Existence of global attractor
Theorem 4.2
Under assumptions(1)–(4), then problems (6)–(9) have a global attractor in\(\mathcal{H}\), which is invariant and compact.
Proof
Lemma 3.1 and Theorem 4.1 imply the existence of the global attractor. □
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The project is supported by the National Natural Science Foundation of China (Grant No. 11872264, the role of the funding lies in the collection of data, the analysis of the paper and writing the manuscript).
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This paper is mainly completed by SZ, JZ and HW dealt with the nonlinear damping term as proving the existence of a bounded absorbing set. All authors read and approved the final manuscript.
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Zhang, S., Zhang, J. & Wang, H. Global attractors for Kirchhoff wave equation with nonlinear damping and memory. Bound Value Probl 2020, 116 (2020). https://doi.org/10.1186/s13661-020-01413-5
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DOI: https://doi.org/10.1186/s13661-020-01413-5