September  2018, 38(9): 4767-4817. doi: 10.3934/dcds.2018210

Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise

1. 

School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

2. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Shengfan Zhou

Received  September 2016 Revised  June 2017 Published  June 2018

Fund Project: This work is supported by the National Natural Science Foundation of China (under Grant Nos. 11471290, 11326114, 11401244) and Natural Science Research Project of Ordinary Universities in Jiangsu Province (grant No. 14KJB110003)

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

Citation: Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
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show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992. Google Scholar

[3]

F. BalibreaT. CaraballoP. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives, Inter. J. Bifur. Chaos, 20 (2010), 2591-2636. doi: 10.1142/S0218127410027246. Google Scholar

[4]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017. Google Scholar

[5]

M. D. BlairH. F. Smith and C. D. Sogge, Strichartz estimates for the wave equation on manifolds with boundary, Ann. Inst. H.Poincaré Anal. Non Lineaire, 26 (2009), 1817-1829. doi: 10.1016/j.anihpc.2008.12.004. Google Scholar

[6]

T. CaraballoJ. A. Langa and J. C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise, Discrete Contin. Dyn. Syst., 6 (2000), 875-892. doi: 10.3934/dcds.2000.6.875. Google Scholar

[7]

T. CaraballoP. E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207. doi: 10.1007/s00245-004-0802-1. Google Scholar

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[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. Google Scholar

[10]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002. doi: 10.1007/b83277. Google Scholar

[11]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. Google Scholar

[12]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474. doi: 10.1023/A:1022605313961. Google Scholar

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H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225. Google Scholar

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A. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Comm. Pure Appl. Anal., 12 (2013), 3047-3071. doi: 10.3934/cpaa.2013.12.3047. Google Scholar

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[16]

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[17]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part Ⅱ: Applications to reaction-diffusion systems, J. Math. Anal. Appl., 381 (2011), 766-780. doi: 10.1016/j.jmaa.2011.03.052. Google Scholar

[18]

A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491. doi: 10.1080/07362999708809490. Google Scholar

[19]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988. doi: 10.1016/S0021-7824(99)80001-4. Google Scholar

[20]

M. EfendievY. Yamamoto and A. Yagi, Exponential attractors for non-autonomous dissipative system, J. Math. Soc. Japan, 63 (2011), 647-673. doi: 10.2969/jmsj/06320647. Google Scholar

[21]

X. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76. doi: 10.2140/pjm.2004.216.63. Google Scholar

[22]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437. doi: 10.1142/S0129167X08004741. Google Scholar

[23]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793. doi: 10.1080/07362990600751860. Google Scholar

[24]

E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Commun.Partial Differential Equations, 18 (1993), 1539-1555. doi: 10.1080/03605309308820985. Google Scholar

[25]

C. Foias and E. Olson, Finite fractal dimension and Holder-Lipschitz parametrization, Indiana Univ. Math. J., 45 (1996), 603-616. doi: 10.1512/iumj.1996.45.1326. Google Scholar

[26]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1007/978-3-642-86458-2_14. Google Scholar

[27]

Y. HuangY. Zhao and Z. Yin, On the dimension of the global attractor for a damped semilinear wave equation with critical exponent, J. Math. Phys., 41 (2000), 4957-4966. doi: 10.1063/1.533386. Google Scholar

[28]

P. Imkeller and B. Schmalfuss, The conjugacy of stochastic and random differential equations and the existence of global attractors, J. Dynam. Differential Equations, 13 (2001), 215-249. doi: 10.1023/A:1016673307045. Google Scholar

[29]

T. JordanM. Pollicott and K. Simon, Hausdorff dimension for randomly perturbed self affine attractors, Commun. Math. Phys., 270 (2006), 519-544. doi: 10.1007/s00220-006-0161-7. Google Scholar

[30]

Y. Kifer, Attractors via random perturbations, Commun. Math. Phys., 121 (1989), 445-455. doi: 10.1007/BF01217733. Google Scholar

[31]

S. Kuksin and A. Shirikyan, Stochastic dissipative PDE's and Gibbs measures, Commun. Math. Phys., 213 (2000), 291-330. doi: 10.1007/s002200000237. Google Scholar

[32]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418. Google Scholar

[33]

J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems, Dyn. Syst., 18 (2003), 57-68. doi: 10.1080/1468936031000080812. Google Scholar

[34]

J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294. doi: 10.1016/j.matpur.2005.08.001. Google Scholar

[35]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discrete Contin. Dyn. Syst., 26 (2010), 1329-1357. doi: 10.3934/dcds.2010.26.1329. Google Scholar

[36]

H. LiY. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190. doi: 10.1016/j.jde.2014.09.007. Google Scholar

[37]

P. Li and S. T. Yau, Estimate of the first eigenvalue of a compact Riemann manifold, Proceeding of Symposition in Pure Math., 36 (1980), 205-239. Google Scholar

[38]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009. Google Scholar

[39]

J. Milnor, On the concept of attractor, Commun. Math. Phys., 99 (1985), 177-195. doi: 10.1007/BF01212280. Google Scholar

[40]

A. MiranvilleV. Pata and S. Zelik, Exponential attractors for singularly perturbed damped wave equations: A simple construction, Asymptot. Anal., 53 (2007), 1-12. Google Scholar

[41]

H. E. Nusse and J. A. Yorke, The equality of fractal dimension and uncertainty dimension for certain dynamical systems, Commun. Math. Phys., 150 (1992), 1-21. doi: 10.1007/BF02096562. Google Scholar

[42]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar

[43]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar

[44]

J. C. Robinson, Stability of random attractors under perturbation and approximation, J. Differential Equations, 186 (2002), 652-669. doi: 10.1016/S0022-0396(02)00038-4. Google Scholar

[45]

D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors, Commun. Math. Phys., 82 (1981/82), 137-151. doi: 10.1007/BF01206949. Google Scholar

[46]

D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Commum. Math. Phys., 93 (1984), 285-300. doi: 10.1007/BF01258529. Google Scholar

[47]

T. SauerJ. A. Yorke and M. Casdagli, Embedology, J. Stat. Phys., 65 (1993), 579-616. doi: 10.1007/BF01053745. Google Scholar

[48]

A. Savostianov and S. Zelik, Recent progress in attractors for quintic wave equations, Math. Bohem., 139 (2014), 657-665. Google Scholar

[49]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. PDE: Anal. Comp., 1 (2013), 241-281. doi: 10.1007/s40072-013-0007-1. Google Scholar

[50]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. Google Scholar

[51]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. Google Scholar

[52]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar

[53]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. Google Scholar

[54]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5. Google Scholar

[55]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stochastics and Dynamics, 14 (2014), 1450009, 31 pp. doi: 10.1142/s0219493714500099. Google Scholar

[56]

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