Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb {R}^3$
HTML articles powered by AMS MathViewer
- by Bixiang Wang PDF
- Trans. Amer. Math. Soc. 363 (2011), 3639-3663 Request permission
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
Additional Information
- Bixiang Wang
- Affiliation: Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801
- MR Author ID: 314148
- Email: bwang@nmt.edu
- Received by editor(s): May 1, 2009
- Received by editor(s) in revised form: November 2, 2009
- Published electronically: February 3, 2011
- Additional Notes: The author was supported in part by NSF grant DMS-0703521
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 3639-3663
- MSC (2000): Primary 37L55; Secondary 60H15, 35B40
- DOI: https://doi.org/10.1090/S0002-9947-2011-05247-5
- MathSciNet review: 2775822