Remark on global attractor for damped wave equation on \(\mathbb{R}^{3}\)
- Fengjuan Meng^{1}Email author and
- Cuncai Liu^{1}
https://doi.org/10.1186/s13662-015-0717-y
© Meng and Liu 2015
Received: 9 September 2015
Accepted: 1 December 2015
Published: 12 December 2015
Abstract
This paper is concerned with the long-time behavior of solutions to the weakly damped wave equation with lower regular forcing defined on the entire space \(\mathbb{R}^{3}\). The existence of a global attractor in \(H^{1}(\mathbb{R}^{3}) \times L^{2}(\mathbb{R}^{3})\) is proven. Moreover, under some additional condition, the translational regularity of the attractor is established.
Keywords
MSC
1 Introduction
On the other hand, in terms of the forcing term g, all papers we mentioned above require \(g\in L^{2}\) or specially \(g=0\). In the case of lower regular forcing, we refer the reader to [20–22], in which the existence and asymptotic regularity of global attractor have been discussed for the strongly damped wave equations. As mentioned in [22], a strongly damped wave equation contains the strong damping term \(-\Delta u_{t}\), which brings about many advantages for considering the long-time behavior, especially in considering the attractor. However, for the weakly damped wave equation (1.1), it seems difficult to apply the corresponding method to verify asymptotic compactness of the solution semigroup. Just recently, in [23], we presented a new method for the decomposition of the solution to deal with the case of \(g\in H^{-1}(\Omega)\) in the bounded domain. Little seems to be known about the weakly damped wave equation in an unbounded domain with lower regular forcing.
The aim of this note is to study the existence and translational regularity of a global attractor for the weakly damped wave equation with lower regular forcing in unbounded domain. It is worth mentioning that the concept of translational regularity of global attractor, in other words, the global attractor being regular after a translation transformation, was introduced in our previous work [23].
Note that the number 2 in (1.2) corresponds to the ‘critical exponent’ case in bounded domains (see [1]), however, in an unbounded domain, it is possible to relax the growth restriction; see [24–30] for the well-posedness of the wave equation without damping and [19, 31] for the existence of global attractor. Due to the lower regular forcing considered in this paper, the well-posedness of problem (1.1) with faster growth nonlinearity is less clear, we shall discuss the weakly damped wave equation with lower regular forcing in unbounded domain when the growth exponent is larger that 3 in a future paper.
The rest of the paper is organized as follows. In Section 2, well-posedness and dissipativity for (1.1) are given. Section 3 is devoted to the compactness of the semigroup and the existence of global attractor, where the proof of compactness for the semigroup is with the help of a decomposition of the solution which inspired by [18, 19, 23]. Finally, the translational regularity of the global attractor in \(\mathcal{H}\) is established in Section 4.
As regards the notations, denote \(H^{\sigma}=\operatorname{dom}((-\Delta)^{\frac{\sigma }{2}})\), then we define the energy product spaces \(\mathcal{H}^{\sigma}=H^{1+\sigma}\times H^{\sigma}\) with \(\mathcal{H}=\mathcal{H}^{0}\) for short, and we write \(\|\cdot\|\) for the norm of \(L^{2}\). Denote by C any positive constant which may be different from line to line or even on the same line, and we also denote the different positive constants by \(C_{i}\), \(i\in\mathbb{N}\), for distinguishing.
2 Well-posedness and dissipativity
Note that \(u_{tt}+\gamma u_{t}-\Delta u-g=-f(x,u) \in L^{2}\), it allows us to multiply \(u_{t}+\alpha u\) over the equation (1.1), and by proper energy estimates (see, e.g., [18, 20]), we can obtain the existence of an absorbing set. We only state the result.
Lemma 2.1
3 Compactness and existence of global attractor
To prove the existence of global attractor by means of well-known results of the theory of dynamical systems, we also need to verify the asymptotic compactness of the solution semigroup \(\{S(t)\}_{t\ge0}\).
Lemma 3.1
Suppose \(g\in H^{-1}(\mathbb{R}^{3})\) and f satisfies (1.2), (1.3), then equation (3.2) has a solution \(h\in H^{1}(\mathbb{R}^{3})\).
Proof
According to assumptions (1.2) and (1.3), the energy functional corresponding to elliptic equation (3.2) is weakly lower semi-continuous and bounded from below on \(H^{1}(\mathbb{R}^{3})\), thus the existence of h can be guaranteed. □
Lemma 3.2
Proof
Proposition 3.3
Lemma 3.4
Proof
Lemmas 3.1, 3.2 and 3.4 show that the solution \((u(t),u_{t}(t))\) with initial data \((u(0),u_{t}(0))\in\mathcal{H}\) decomposes into the sum of a fixed point \((h,0)\) and a uniform decaying term \((v(t),v_{t}(t))\) and a term \((w(t),w_{t}(t))\) belonging to a compact set \(\mathfrak{C}\subset\mathcal{H}\). Therefore, we can derive the asymptotic compactness of the semigroup \(\{S(t)\}_{t\geq0} \) immediately. Combining with the existence of a bounded absorbing set stated in Lemma 2.1, the existence of a global attractor is obtained by the standard methods of the theory of attractors (see, e.g., [7, 11, 34]). We state the result.
Theorem 3.5
Let \(g\in H^{-1}(\mathbb{R}^{3})\) and \(f\in C^{1}(\mathbb{R}^{4})\) satisfy (1.2) and (1.3), \(\{S(t)\}_{t\geq0}\) be the semigroup generated by the solution of problem (1.1) in \(\mathcal{H}\). Then \(\{S(t)\}_{t\geq0}\) possesses a global attractor \(\mathscr{A}\) in \(\mathcal{H}\), which is compact and invariant and attracts the bounded sets of \(\mathcal{H}\).
4 Translational regularity of global attractor
Due to the solution of the corresponding stationary equation of (1.1) \(-\Delta h+\lambda h+f(h)=g\) only belonging to \(H^{1}\), we cannot expect any higher regularity of the global attractor than \(\mathcal {H}\). In [23], we obtained the translational regularity of a global attractor for the bounded domain. For the unbounded domain in the present text, if we add some condition on f other than (1.2) and (1.3), we can also obtain the translational regularity of the global attractor.
Since equation (4.2) is linear, it is easy to check that \(\tilde {v}(t)\) is exponentially decaying. For the w̃-component, we have the following regularity lifting lemma.
Lemma 4.1
Note that \(\mathscr{A}=\bigcap_{t\ge0}\overline{\bigcup_{s \ge t}S(s)\mathscr{B}_{0}}\). The decaying property of ṽ implies that \(\mathscr{A}\subset(h,0)+\bigcap_{t\ge0}\overline{\bigcup_{s \ge t}S_{\tilde{w}}(s)\mathscr{B}_{0}}\). Applying Lemma 4.1, we have the following translational regularity of the global attractor.
Declarations
Acknowledgements
This work was supported by Natural Science Fund For Colleges and Universities in Jiangsu Province (15KJB110005), TianYuan Special Funds of the National Natural Science Foundation of China and the Foundation of Jiangsu University of Technology (KYY14045).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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