Asymptotic behavior of stochastic wave equations with critical exponents on ℝ³

Wang, Bixiang

doi:10.1090/S0002-9947-2011-05247-5

Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb {R}^3$
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by Bixiang Wang

Trans. Amer. Math. Soc. 363 (2011), 3639-3663

DOI: https://doi.org/10.1090/S0002-9947-2011-05247-5

Published electronically: February 3, 2011

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Abstract:

The existence of a random attractor in $H^1(\mathbb {R}^3) \times L^2(\mathbb {R}^3)$ is proved for the damped semilinear stochastic wave equation defined on the entire space $\mathbb {R}^3$. The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method.

References

Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
José Arrieta, Alexandre N. Carvalho, and Jack K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations 17 (1992), no. 5-6, 841–866. MR 1177295, DOI 10.1080/03605309208820866
A. V. Babin and M. I. Vishik, Attractors of evolution equations, Studies in Mathematics and its Applications, vol. 25, North-Holland Publishing Co., Amsterdam, 1992. Translated and revised from the 1989 Russian original by Babin. MR 1156492
J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 31–52. Partial differential equations and applications. MR 2026182, DOI 10.3934/dcds.2004.10.31
J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), no. 5, 475–502. MR 1462276, DOI 10.1007/s003329900037
Peter W. Bates, Hannelore Lisei, and Kening Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn. 6 (2006), no. 1, 1–21. MR 2210679, DOI 10.1142/S0219493706001621
Peter W. Bates, Kening Lu, and Bixiang Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations 246 (2009), no. 2, 845–869. MR 2468738, DOI 10.1016/j.jde.2008.05.017
Zdzisław Brzeźniak and Yuhong Li, Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5587–5629. MR 2238928, DOI 10.1090/S0002-9947-06-03923-7
Tomás Caraballo, José A. Langa, and James C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2013, 2041–2061. MR 1857922, DOI 10.1098/rspa.2001.0819
T. Caraballo, J. Real, and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), no. 3-4, 525–539. MR 2379425, DOI 10.3934/dcdsb.2008.9.525
Igor Chueshov and Irena Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations 16 (2004), no. 2, 469–512. MR 2105786, DOI 10.1007/s10884-004-4289-x
Igor Chueshov and Michael Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst. 19 (2004), no. 2, 127–144. MR 2060422, DOI 10.1080/1468936042000207792
Hans Crauel, Arnaud Debussche, and Franco Flandoli, Random attractors, J. Dynam. Differential Equations 9 (1997), no. 2, 307–341. MR 1451294, DOI 10.1007/BF02219225
Hans Crauel and Franco Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields 100 (1994), no. 3, 365–393. MR 1305587, DOI 10.1007/BF01193705
Giuseppe Da Prato, An introduction to infinite-dimensional analysis, Universitext, Springer-Verlag, Berlin, 2006. Revised and extended from the 2001 original by Da Prato. MR 2244975, DOI 10.1007/3-540-29021-4
Eduard Feireisl, Attractors for semilinear damped wave equations on $\textbf {R}^3$, Nonlinear Anal. 23 (1994), no. 2, 187–195. MR 1289126, DOI 10.1016/0362-546X(94)90041-8
Eduard Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations 116 (1995), no. 2, 431–447. MR 1318582, DOI 10.1006/jdeq.1995.1042
Eduard Feireisl and Enrique Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1539–1555. MR 1239923, DOI 10.1080/03605309308820985
Franco Flandoli and Björn Schmalfuss, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep. 59 (1996), no. 1-2, 21–45. MR 1427258, DOI 10.1080/17442509608834083
Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
Alain Haraux, Semi-linear hyperbolic problems in bounded domains, Math. Rep. 3 (1987), no. 1, i–xxiv and 1–281. MR 1078761
A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations 230 (2006), no. 2, 702–719. MR 2269940, DOI 10.1016/j.jde.2006.06.001
Peter E. Kloeden and José A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2077, 163–181. MR 2281716, DOI 10.1098/rspa.2006.1753
Ning Ju, The $H^1$-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains, Nonlinearity 13 (2000), no. 4, 1227–1238. MR 1767956, DOI 10.1088/0951-7715/13/4/313
Ioana Moise, Ricardo Rosa, and Xiaoming Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity 11 (1998), no. 5, 1369–1393. MR 1644413, DOI 10.1088/0951-7715/11/5/012
Ioana Moise, Ricardo Rosa, and Xiaoming Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 473–496. Partial differential equations and applications. MR 2026206, DOI 10.3934/dcds.2004.10.473
Martino Prizzi and Krzysztof P. Rybakowski, Attractors for semilinear damped wave equations on arbitrary unbounded domains, Topol. Methods Nonlinear Anal. 31 (2008), no. 1, 49–82. MR 2420655
Martino Prizzi and Krzysztof P. Rybakowski, Attractors for singularly perturbed damped wave equations on unbounded domains, Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 1–20. MR 2466799
Martino Prizzi, Regularity of invariant sets in semilinear damped wave equations, J. Differential Equations 247 (2009), no. 12, 3315–3337. MR 2571579, DOI 10.1016/j.jde.2009.08.011
George R. Sell and Yuncheng You, Dynamics of evolutionary equations, Applied Mathematical Sciences, vol. 143, Springer-Verlag, New York, 2002. MR 1873467, DOI 10.1007/978-1-4757-5037-9
Yan Guo (ed.), Nonlinear wave equations, Contemporary Mathematics, vol. 263, American Mathematical Society, Providence, RI, 2000. Papers from the conference in honor of Walter A. Strauss on the occasion of his 60th birthday held at Brown University, Providence, RI, May 2–3, 1998. MR 1777631, DOI 10.1090/conm/263
Chunyou Sun, Daomin Cao, and Jinqiao Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity 19 (2006), no. 11, 2645–2665. MR 2267722, DOI 10.1088/0951-7715/19/11/008
Chunyou Sun, Meihua Yang, and Chengkui Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations 227 (2006), no. 2, 427–443. MR 2237675, DOI 10.1016/j.jde.2005.09.010
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312, DOI 10.1007/978-1-4612-0645-3
Bixiang Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D 128 (1999), no. 1, 41–52. MR 1685247, DOI 10.1016/S0167-2789(98)00304-2
Bixiang Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations 246 (2009), no. 6, 2506–2537. MR 2498851, DOI 10.1016/j.jde.2008.10.012
Bixiang Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. 71 (2009), no. 7-8, 2811–2828. MR 2532807, DOI 10.1016/j.na.2009.01.131
Bixiang Wang and Xiaoling Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst. Dynamical systems, differential equations and applications. 7th AIMS Conference, suppl. (2009), 800–809. MR 2648205, DOI 10.1016/j.nonrwa.2011.06.008
Bixiang Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations (2009), No. 139, 18. MR 2558811
Xiaoming Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Phys. D 88 (1995), no. 3-4, 167–175. MR 1360882, DOI 10.1016/0167-2789(95)00196-B
Shengfan Zhou, Fuqi Yin, and Zigen Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst. 4 (2005), no. 4, 883–903. MR 2179491, DOI 10.1137/050623097