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


Author: Bixiang Wang
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
Published electronically: February 3, 2011
MathSciNet review: 2775822
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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.


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  • 1. L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998. MR 1723992 (2000m:37087)
  • 2. J.M. Arrieta, A.N. Carvalho and J.K. Hale, A damped hyperbolic equation with critical exponent, Communications in Partial Differential Equations, 17 (1992), 841-866. MR 1177295 (93f:35145)
  • 3. A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. MR 1156492 (93d:58090)
  • 4. J.M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynamical Systems, 10 (2004), 31-52. MR 2026182 (2005a:37149)
  • 5. J.M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7 (1997), 475-502. MR 1462276 (98j:58071a)
  • 6. P.W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. MR 2210679 (2006m:60077)
  • 7. P.W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. MR 2468738 (2009i:35357)
  • 8. Z. Brzezniak and Y. Li, Asymptotic compactness and absorbing sets for 2d stochastic Navier-Stokes equations on some unbounded domains, Transactions of American Math. Soc., 358 (2006), 5587-5629. MR 2238928 (2008i:60101)
  • 9. T. Caraballo, J. A. Langa and J. C. Robinson, A stochastic pitchfork bifurcation in a reaction-diffusion equation, Proc. R. Soc. Lond. A, 457 (2001), 2041-2061. MR 1857922 (2003a:60106)
  • 10. T. Caraballo, J. Real, I.D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Continuous Dynamical Systems B, 9 (2008), 525-539. MR 2379425 (2008m:60111)
  • 11. I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. MR 2105786 (2005g:37149)
  • 12. I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dynamical Systems, 19 (2004), 127-144. MR 2060422 (2005c:37095)
  • 13. H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. MR 1451294 (98c:60066)
  • 14. H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. MR 1305587 (95k:58092)
  • 15. G. Da Prato, An Introduction to Infinite-Dimensional Analysis, Springer-Verlag, Berlin, 2006. MR 2244975 (2009a:46001)
  • 16. E. Feireisl, Attractors for semilinear damped wave equations on $ \mathbb{R}^3$, Nonlinear Analysis TMA, 23 (1994), 187-195. MR 1289126 (95i:35188)
  • 17. E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995) 431-447. MR 1318582 (96a:35120)
  • 18. E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent, Communications in Partial Differential Equations 18 (1993) 1539-1555. MR 1239923 (95c:35172)
  • 19. F. Flandoli and B. Schmalfu$ \beta$, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45. MR 1427258 (98g:60113)
  • 20. J.K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. MR 941371 (89g:58059)
  • 21. A. Haraux, Semi-Linear Hyperbolic Problems in Bounded Domains, Mathematical Reports, Vol. 3, Part 1, Harwood Academic Publishers, New York, 1987. MR 1078761 (91m:35150)
  • 22. A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719. MR 2269940 (2007i:37150)
  • 23. P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Series A., 463 (2007), 163-181. MR 2281716 (2008a:37056)
  • 24. N. Ju, The $ H^1$-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains, Nonlinearity, 13 (2000), 1227-1238. MR 1767956 (2001f:37131)
  • 25. I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393. MR 1644413 (99h:35020)
  • 26. I. Moise, R. Rosa and X. Wang, Attractors for non-compact nonautonomous systems via energy equations, Discrete Continuous Dynamical Systems, 10 (2004), 473-496. MR 2026206 (2004m:37151)
  • 27. M. Prizzi and K.P. Rybakowski, Attractors for semilinear damped wave equations on arbitrary unbounded domains, Topol. Methods Nonlinear Anal., 31 (2008),49-82. MR 2420655 (2009h:35287)
  • 28. M. Prizzi and K. P. Rybakowski, Attractors for singularly perturbed damped wave equations on unbounded domains, Topol. Methods Nonlinear Anal., 32 (2008), 1-20. MR 2466799
  • 29. M. Prizzi, Regularity of invariant sets in semilinear damped wave equations, J. Differential Equations, 247 (2009), 3315-3337. MR 2571579 (2010k:35317)
  • 30. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. MR 1873467 (2003f:37001b)
  • 31. W.A. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series in Mathematics, Vol. 73, American Mathematical Society, 1990. MR 1777631 (2001c:35002)
  • 32. C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665. MR 2267722 (2007h:35235)
  • 33. C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, J. Differential Equations, 227 (2006), 427-443. MR 2237675 (2007j:37140)
  • 34. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. MR 1441312 (98b:58056)
  • 35. B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. MR 1685247 (2000a:35126)
  • 36. B. Wang, Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains, J. Differential Equations, 246 (2009), 2506-2537. MR 2498851
  • 37. B. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Analysis, TMA, 71 (2009), 2811-2828. MR 2532807
  • 38. B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Continuous Dynamical Systems, special (2009), 800-809. MR 2648205
  • 39. B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), No. 139, 1-18. MR 2558811
  • 40. X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications, Physica D, 88 (1995), 167-175. MR 1360882 (96h:35215)
  • 41. S. Zhou, F. Yin and Z. Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903. MR 2179491 (2006g:60096)

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Additional Information

Bixiang Wang
Affiliation: Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801
Email: bwang@nmt.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05247-5
Keywords: Random attractor, asymptotic compactness, wave equation.
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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