- Open Access
Upper semicontinuity of attractors for small perturbations of Klein-Gordon-Schrödinger lattice system
© Li and Sun; licensee Springer. 2014
Received: 4 October 2014
Accepted: 17 November 2014
Published: 1 December 2014
This paper studies the existence of the random attractor for a Klein-Gordon-Schödinger system under a small ε-random perturbation on a high dimensional infinite lattice. Firstly, we prove the asymptotic compactness of the random dynamical system and obtain the random attractor. Then, by comparing to the case without random perturbation (), we show the upper semicontinuity of the attractors.
where , , , ℤ is the integer set, α, μ, and ε are positive constants, , , and , , , the space of bounded continuous functions from ℝ into . and are two independent two-side real valued standard Wiener processes, linear operator A and space will be described in detail in the next section.
where u denotes a complex scalar nucleon field and v represents a real meson field; the complex-valued function and the real-valued function both are the time-dependent external sources. By using the Galerkin method, Fukuda and Tsutsumi  first studied the coupled KGS equations, and they obtained the existence of global strong solutions. We refer the readers to [2–7] for the existence of global solutions, asymptotic behavior, and stability for the KGS system and KGS system on infinite lattices.
where denotes the Hausdorff semidistance.
This paper is organized as follows. In Section 2, we recall some basic concepts and already known results related to random dynamical systems and random attractors. In Section 3, we prove the existence of the random attractor for stochastic KGS lattice dynamical systems of (1.1) on . In Section 4, by comparing to the case without the random perturbation, i.e. case, we obtain the upper semicontinuity of the attractors.
Let be a Hilbert space, be a probability space.
Definition is called a metric dynamical systems, if is -measurable, , for all , and for all .
Definition A stochastic process is called a continuous random dynamical system (RDS) over if ϕ is -measurable, and for all
the mapping is continuous;
for all and (cocycle property).
Consider a continuous random dynamical system over and let be the collection of all tempered random set of H.
Definition 2.3 A random set is a random -attractor for RDS ϕ if
is a random compact set, i.e., is measurable for every and is compact for a.e. ;
is strictly invariant, i.e., , and for a.e. ;
attracts all sets in , i.e., for all and a.e. we have
where , .
The collection is called the domain of attraction of .
Definition 2.4 Let ϕ be a RDS on Hilbert space H. ϕ is called asymptotically compact if for any bounded sequence and , the set is precompact in H, for any .
From , we have the following result.
which is compact in H.
where denotes the conjugate of .
Let and , where . Here denotes the standard complete orthonormal system in , which means that the j th component of is 1 and all other elements are 0. Then and are -valued Q-Wiener processes. It is obvious that and . For details we refer to .
endowed with the compact open topology . ℙ is the corresponding Wiener measure and is the ℙ-completion of the Borel σ-algebra on Ω.
Let , . Then is a metric dynamical system with the filtration , , where is the smallest σ-algebra generated by the random variable for all , such that ; see  for more details.
3 The existence of a random attractor
For each fixed , system (3.3) is a deterministic equation, and we have the following result.
Theorem 3.1 For any , system (2.2) is well-posed and admits a unique solution . Moreover, the solution of (2.2) depends continuously on the initial data .
where , are constants depending on α, σ, μ, ε, and . By the Gaussian property of and , (3.4) implied that system (3.3) admits the global solution . The proof is completed. □
and combining the above theorem we have the following result.
Theorem 3.2 System (2.2) generates a continuous random dynamical system over .
The proof is similar to that of Theorem 3.2 in , so we omit it.
the mappings , , are continuous for each ;
the random variables , , are tempered.
Lemma 3.2 There exists a invariant set of full ℙ measure and an absorbing random set , , for the random dynamical system .
Then is a tempered ball by the property of , , and, for any , . Here, denotes the collection of all tempered random set of Hilbert space H. This completes the proof. □
where for .
and (a positive constant).
This completes the proof. □
Lemma 3.4 The random dynamical system is asymptotically compact.
Next, we show that the above weak convergence is actually strong convergence in E.
This completes the proof. □
Now, combining Lemma 3.2, Lemma 3.4, and Proposition 2.1, we can easily obtain the following result.
Theorem 3.3 The random dynamical system possesses a global random attractor in E.
4 Upper semicontinuity of attractors
This section studies the upper semicontinuity of random attractors for the stochastic Klein-Gordon-Schrödinger lattice system. The existence of global attractors for the Klein-Gordon-Schrödinger lattice system has been obtained by  for one dimension, and  for the high dimensional case. We assume is an attractor corresponding to the Klein-Gordon-Schrödinger lattice system, i.e., the case of system (1.1). By Theorem 2 of , we only need to prove the following lemma for upper semicontinuity of attractors. In what follows, we take the vector form for brevity.
where is a positive constant, and .
where , are constants depending on α, σ, μ, n, C, and .
This completes the proof. □
The authors express their sincere thanks to the anonymous referees for a very careful reading and for providing many valuable comments and suggestions, which led to an improvement of this paper. The first author is supported by the Fundamental Research Foundation for PHD of NCWU (No. 40291).
- Fukuda I, Tsutsumi M: On the coupled Klein-Gordon-Schrödinger equations, III. Math. Jpn. 1979, 24: 307-321.MathSciNetGoogle Scholar
- Guo B, Li Y:Attractor for dissipative Klein-Gordon-Schrödinger equations in . J. Differ. Equ. 1997, 136: 356-377. 10.1006/jdeq.1996.3242View ArticleGoogle Scholar
- Wang BX, Lange H: Attractors for the Klein-Gordon-Schrödinger equation. J. Math. Phys. 1999, 40(5):2445-2457. 10.1063/1.532875MathSciNetView ArticleGoogle Scholar
- Lu KN, Wang BX: Global attractors for the Klein-Gordon-Schrödinger equations in unbounded domains. J. Differ. Equ. 2001, 170: 281-361. 10.1006/jdeq.2000.3827View ArticleGoogle Scholar
- Biler P: Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling. SIAM J. Math. Anal. 1990, 21(5):1190-1212. 10.1137/0521065MathSciNetView ArticleGoogle Scholar
- Zhao CD, Zhou SF: Compact kernel sections for nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices. J. Math. Anal. Appl. 2007, 332: 32-56. 10.1016/j.jmaa.2006.10.002MathSciNetView ArticleGoogle Scholar
- Lu KN, Wang BX: Upper semicontinuity of attractors for the Klein-Gordon-Schrödinger equation. Int. J. Bifurc. Chaos Appl. Sci. Eng. 2005, 15(1):157-168. 10.1142/S0218127405012077View ArticleGoogle Scholar
- Temam R: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York; 1995.Google Scholar
- Bates PW, Lu KN, Wang BX: Attractors for lattice dynamical systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 2001, 11(1):143-153. 10.1142/S0218127401002031MathSciNetView ArticleGoogle Scholar
- Chow SN: Lattice dynamical systems. Lect. Notes in Math. 1822. Dynamical Systems 2003, 1-102.View ArticleGoogle Scholar
- Chow SN, Paret JM: Pattern formation and spatial chaos in lattice dynamical systems. IEEE Trans. Circuits Syst. 1995, 42: 746-751. 10.1109/81.473583View ArticleGoogle Scholar
- Fan XM, Wang YG: Attractors for a second order nonautonomous lattice dynamical system with nonlinear damping. Phys. Lett. A 2007, 365: 17-27. 10.1016/j.physleta.2006.12.045MathSciNetView ArticleGoogle Scholar
- Wang BX: Attractors for reaction-diffusion equations in unbounded domains. Physica D 1999, 128: 41-52. 10.1016/S0167-2789(98)00304-2MathSciNetView ArticleGoogle Scholar
- Wang BX: Dynamics of lattice systems on infinite lattices. J. Differ. Equ. 2006, 221: 224-245. 10.1016/j.jde.2005.01.003View ArticleGoogle Scholar
- Abdallah AY: Long-time behavior for second order lattice dynamical systems. Acta Appl. Math. 2008. 10.1007/s10440-008-9281-8Google Scholar
- Abdallah AY: Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Commun. Pure Appl. Anal. 2006, 5(1):55-69.MathSciNetView ArticleGoogle Scholar
- Zhou SF: Attractors for first order dissipative lattice dynamical systems. Physica D 2003, 178: 51-61. 10.1016/S0167-2789(02)00807-2MathSciNetView ArticleGoogle Scholar
- Bates PW, Chen X, Chmaj A: Travelling waves of bistable dynamics on a lattice. SIAM J. Math. Anal. 2003, 35: 520-546. 10.1137/S0036141000374002MathSciNetView ArticleGoogle Scholar
- Chua LO, Roska T: The CNN paradigm. IEEE Trans. Circuits Syst. 1993, 40: 147-156.MathSciNetView ArticleGoogle Scholar
- Kapval R: Discrete models for chemically reacting systems. J. Math. Chem. 1991, 6: 113-163. 10.1007/BF01192578MathSciNetView ArticleGoogle Scholar
- Chow SN, Paret JM, Vleck ES: Pattern formation and spatial chaos in spatially discrete evolution equations. Random Comput. Dyn. 1996, 4: 109-178.Google Scholar
- Carrol TL, Pecora LM: Synchronization in chaotic systems. Phys. Rev. Lett. 1990, 64: 821-824. 10.1103/PhysRevLett.64.821MathSciNetView ArticleGoogle Scholar
- Fabiny L, Colet P: Coherence and phase dynamics of spatially coupled solid-state lasers. Phys. Rev. A 1993, 47: 4287-4296. 10.1103/PhysRevA.47.4287View ArticleGoogle Scholar
- Bates PW, Lisei H, Lu KN: Attractors for stochastic lattice dynamical systems. Stoch. Dyn. 2006, 6: 1-21. 10.1142/S0219493706001621MathSciNetView ArticleGoogle Scholar
- Lv Y, Sun J: Dynamical behavior for stochastic lattice systems. Chaos Solitons Fractals 2006, 27: 1080-1090. 10.1016/j.chaos.2005.04.089MathSciNetView ArticleGoogle Scholar
- Lv Y, Sun J: Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations. Physica D 2006, 221: 157-169. 10.1016/j.physd.2006.07.023MathSciNetView ArticleGoogle Scholar
- Huang JH: The random attractor of stochastic Fitzburg-Nagumo equations in an infinite lattice with white noises. Physica D 2007, 233: 83-94. 10.1016/j.physd.2007.06.008MathSciNetView ArticleGoogle Scholar
- Ruelle D: Characteristic exponents for a viscous fluid subjected to time dependent forces. Commun. Math. Phys. 1984, 93: 285-300. 10.1007/BF01258529MathSciNetView ArticleGoogle Scholar
- Arnold L: Random Dynamical Systems. Springer, Berlin; 1998.View ArticleGoogle Scholar
- Crauel H, Debussche A, Flandoli F: Random attractors. J. Dyn. Differ. Equ. 1997, 9: 307-341. 10.1007/BF02219225MathSciNetView ArticleGoogle Scholar
- Yan WP, Ji SG, Li Y: Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations. Phys. Lett. A 2009, 373: 1268-1275. 10.1016/j.physleta.2009.02.019MathSciNetView ArticleGoogle Scholar
- Da Prato GD, Zabczyk J London Mathematical Society Lecture Note Series 229. In Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge; 1996.View ArticleGoogle Scholar
- Caraballo T, Langa JA, Robinson C: Upper semicontinuity of attractors for small random perturbations of dynamical systems. Commun. Partial Differ. Equ. 1998, 23: 1557-1581. 10.1080/03605309808821394MathSciNetView ArticleGoogle Scholar
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