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 and Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
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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.

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M. YangJ. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.  doi: 10.1016/j.nonrwa.2010.06.032.

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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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

R. Carmona and D. Nualart, Random nonlinear wave equations:Smoothness of the solutions, Proba. Theory Relat. Fields, 79 (1988), 469-508.  doi: 10.1007/BF00318783.

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[14]

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.

[15]

A. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Comm. Pure Appl. Anal., 13 (2014), 1141-1165.  doi: 10.3934/cpaa.2014.13.1141.

[16]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part Ⅰ: Semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[26]

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

[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.

[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.

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[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.

[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.

[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.

[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. 

[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.

[39]

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

[40]

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

[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.

[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.

[43]

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

[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.

[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.

[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.

[47]

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

[48]

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

[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.

[50]

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

[51]

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

[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.

[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.

[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.

[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.

[56]

G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications, J. Appl. Math., (2013), Art. ID 415764, 5 pp. doi: 10.1155/2013/415764.

[57]

M. YangJ. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.  doi: 10.1016/j.nonrwa.2010.06.032.

[58]

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