July  2020, 25(7): 2793-2824. doi: 10.3934/dcdsb.2020033

Random attractors for stochastic time-dependent damped wave equation with critical exponents

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

* Corresponding author: Chunyou Sun

Received  May 2019 Revised  September 2019 Published  April 2020

We study the asymptotic behavior of solutions of a stochastic time-dependent damped wave equation. With the critical growth restrictions on the nonlinearity $ f $ and the time-dependent damped term, we prove the global existence of solutions and characterize their long-time behavior. We show the existence of random attractors with finite fractal dimension in $ H^1_0(U)\times L^2(U) $. In particular, the periodicity of random attractors is also obtained with periodic force term and coefficient function. Furthermore, we construct the pullback random exponential attractors.

Citation: Qingquan Chang, Dandan Li, Chunyou Sun. Random attractors for stochastic time-dependent damped wave equation with critical exponents. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2793-2824. doi: 10.3934/dcdsb.2020033
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T. Caraballo and S. Sonner, Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 6383-6403.  doi: 10.3934/dcds.2017277.  Google Scholar

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[26]

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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

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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.  Google Scholar

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M. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. Real World Appl., 12 (2011), 2811-2821.  doi: 10.1016/j.nonrwa.2011.04.007.  Google Scholar

[33]

S. Zhou and M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear Anal., 133 (2016), 292-318.  doi: 10.1016/j.na.2015.12.013.  Google Scholar

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Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 545-573.  doi: 10.3934/dcds.2017022.  Google Scholar

show all references

References:
[1]

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

[2]

J. M. Ball, On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations, 27 (1978), 224-265.  doi: 10.1016/0022-0396(78)90032-3.  Google Scholar

[3]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31–52. doi: 10.3934/dcds.2004.10.31.  Google Scholar

[4]

T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A gradient-like nonautonomous evolution process, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2751-2760.  doi: 10.1142/S0218127410027337.  Google Scholar

[5]

T. CaraballoA. N. CarvalhoJ. A. Langa and F. Rivero, A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor, Nonlinear Anal., 74 (2011), 2272-2283.  doi: 10.1016/j.na.2010.11.032.  Google Scholar

[6]

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.   Google Scholar

[7]

T. Caraballo and S. Sonner, Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces, Discrete Contin. Dyn. Syst., 37 (2017), 6383-6403.  doi: 10.3934/dcds.2017277.  Google Scholar

[8]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[9]

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

[10]

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

[11]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[12]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.  Google Scholar

[13]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[14]

I. ChueshovM. Eller and I. Lasiecka, Finite dimensionality of the attractor for a semilinear wave equation with nonlinear boundary dissipation, Comm. Partial Differential Equations, 29 (2004), 1847-1876.  doi: 10.1081/PDE-200040203.  Google Scholar

[15]

I. ChueshovI. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations, 21 (2009), 269-314.  doi: 10.1007/s10884-009-9132-y.  Google Scholar

[16]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22903-4.  Google Scholar

[17]

I. ChueshovP. E. Kloeden and M. Yang, Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 991-1009.  doi: 10.3934/dcdsb.2018139.  Google Scholar

[18]

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

[19]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.  Google Scholar

[20]

L. C. Evens, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[21]

K. P. Hadeler, Reaction Telegraph Equation and Random Walk Systems, Stochastic and Spatial Structures of Dynamical Systems, 161, North Holland, Amsterdam, 1996,133-161.  Google Scholar

[22]

A. Haraux, Sharp estimates of bounded solutions to some second-order forced dissipative equations, J. Dynam. Differential Equations, 19 (2007), 915-933.  doi: 10.1007/s10884-007-9072-3.  Google Scholar

[23]

W. Hayt, Engineering Electromagnetics, McGraw-Hill, 1989. Google Scholar

[24]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, 9, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009.  Google Scholar

[25]

M. Reissig, $L^p$-$L^q$ decay estimates for wave equations with time-dependent coefficients, J. Nonlinear Math. Phys., 11 (2004), 534–548. doi: 10.2991/jnmp.2004.11.4.9.  Google Scholar

[26]

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

[27]

R. Teman, 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

[28]

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

[29]

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

[30]

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

[31]

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.  Google Scholar

[32]

M. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equations, Nonlinear Anal. Real World Appl., 12 (2011), 2811-2821.  doi: 10.1016/j.nonrwa.2011.04.007.  Google Scholar

[33]

S. Zhou and M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear Anal., 133 (2016), 292-318.  doi: 10.1016/j.na.2015.12.013.  Google Scholar

[34]

Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Discrete Contin. Dyn. Syst., 37 (2017), 545-573.  doi: 10.3934/dcds.2017022.  Google Scholar

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