December  2020, 28(4): 1529-1544. doi: 10.3934/era.2020080

Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: liyr@swu.edu.cn (Yangrong Li)

Received  March 2020 Revised  June 2020 Published  July 2020

Fund Project: This work was supported by Natural Science Foundation of China grant 11571283

We study the random dynamics for the stochastic non-autonomous Kuramoto-Sivashinsky equation in the possibly non-dissipative case. We first prove the existence of a pullback attractor in the Lebesgue space of odd functions, then show that the fiber of the odd pullback attractor semi-converges to a nonempty compact set as the time-parameter goes to minus infinity and finally prove the measurability of the attractor. In a word, we obtain a longtime stable random attractor formed from odd functions. A key tool is the existence of a bridge function between Lebesgue and Sobolev spaces of odd functions.

Citation: Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
References:
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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

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[24]

R. Wang and Y. Li, Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4145-4167.  doi: 10.3934/dcdsb.2019054.  Google Scholar

[25]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382/383 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar

[26]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

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show all references

References:
[1]

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

[2]

Z. Brzeźniak and Y. Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Tran. Amer. Math. Soc., 358 (2006), 5587-5629.  doi: 10.1090/S0002-9947-06-03923-7.  Google Scholar

[3]

T. CaraballoA. N. CarvalhoH. B. da Costa and J. A. Langa, Equi-attraction and continuity of attractors for skew-product semiflows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2949-2967.  doi: 10.3934/dcdsb.2016081.  Google Scholar

[4]

H. CrauelP. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[5]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar

[6]

H. CuiJ. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

[7]

P. Gao, Averaging Principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation, Discrete Contin. Dyn. Syst., 38 (2018), 5649-5684.  doi: 10.3934/dcds.2018247.  Google Scholar

[8]

D. Goluskin and G. Fantuzzi, Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming, Nonlinearity, 32 (2019), 1705-1730.  doi: 10.1088/1361-6544/ab018b.  Google Scholar

[9]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar

[10]

M. HutzenthalerA. Jentzen and D. Salimova, Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations, Commun. Math. Sci., 16 (2018), 1489-1529.  doi: 10.4310/CMS.2018.v16.n6.a2.  Google Scholar

[11]

G. Hwang and B. Moon, Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion, Elect. Research Archive, 28 (2020), 15-25.  doi: 10.3934/era.2020002.  Google Scholar

[12]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[13]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[14]

A. KrauseM. Lewis and B. Wang, Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.  doi: 10.1016/j.amc.2014.08.033.  Google Scholar

[15]

A. N. Kulikov and D. A. Kulikov, The Kuramoto-Sivashinsky equation. A local attractor filled with unstable periodic solutions, Model. Anal. Inf. Sist., 25 (2018), 92-101.  doi: 10.18255/1818-1015-2018-1-92-101.  Google Scholar

[16]

Y. Kuramoto, Diffusion induced chaos in reactions systems, Progr. Theoret. Phys. Suppl., 64 (1978), 346-367.  doi: 10.1143/PTPS.64.346.  Google Scholar

[17]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[18]

Y. Li and S. Yang, Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 1155-1175.  doi: 10.3934/cpaa.2019056.  Google Scholar

[19]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar

[20]

R. K. Mohanty and D. Kaur, High accuracy two-level implicit compact difference scheme for 1D unsteady biharmonic problem of first kind: Application to the generalized Kuramoto-Sivashinsky equation, J. Difference Equ. Appl., 25 (2019), 243-261.  doi: 10.1080/10236198.2019.1568423.  Google Scholar

[21]

B. NicolaenkoB. Scheurer and R. Temam, Some global dynamical properties of a class of pattern formation equations, Commun. Partial Differential Equations, 14 (1989), 245-297.  doi: 10.1080/03605308908820597.  Google Scholar

[22]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[23]

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

[24]

R. Wang and Y. Li, Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4145-4167.  doi: 10.3934/dcdsb.2019054.  Google Scholar

[25]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382/383 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar

[26]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[27]

W. WuS.-B. Cui and J.-Q. Duan, Global well-posedness of the stochastic generalized Kuramoto-Sivashinsky equation with multiplicative noise, Acta Math. Appl. Sin. Engl. Ser., 34 (2018), 566-584.  doi: 10.1007/s10255-018-0769-3.  Google Scholar

[28]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $\Bbb{R}^3$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.  Google Scholar

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