Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity

Authors: Meihua Yang and Chunyou Sun
Journal: Trans. Amer. Math. Soc. 361 (2009), 1069-1101
MSC (2000): Primary 37L05, 35B40, 35B41
Published electronically: September 29, 2008
MathSciNet review: 2452835
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is dedicated to analyzing the dynamical behavior of strongly damped wave equations with critical nonlinearity in locally uniform spaces. After proving the global well-posedness, we first establish the asymptotic regularity of the solutions which appears to be optimal and the existence of a bounded (in $ H^2_{lu}(\mathbb{R}^N)\times H^1_{lu}(\mathbb{R}^N)$) subset which attracts exponentially every initial $ H^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N)$-bounded set with respect to the $ H^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N)$-norm. Then, we show there is a $ (H ^1_{lu}(\mathbb{R}^N)\times L^2_{lu}(\mathbb{R}^N), H^1_\rho(\mathbb{R}^N)\times H^1_\rho(\mathbb{R}^N))$-global attractor, which reflects the strongly damped property of $ \Delta u_t$ to some extent.

References [Enhancements On Off] (What's this?)

  • 1. J. Arrieta, A.N. Carvalho and J.K. Hale, A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17 (1992), 841-866. MR 1177295 (93f:35145)
  • 2. J. Arrieta, J.W. Cholewa, T. Dlotko and A. Rodriguez-Bernal, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. MR 2040897 (2004m:35114)
  • 3. A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. MR 1156492 (93d:58090)
  • 4. A.V. Babin and M.I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243. MR 1084733 (91m:35106)
  • 5. V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $ \mathbb{R}^3$, Discrete Contin. Dyn. Syst., 7 (2001), 719-735. MR 1849655 (2003f:35026)
  • 6. A.N. Carvalho and J.W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310. MR 1972247 (2004b:35023)
  • 7. A.N. Carvalho and J.W. Cholewa, Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66 (2002), 443-463. MR 1939206 (2004b:35228)
  • 8. A.N. Carvalho and T. Dlotko, Partly dissipative systems in uniformly local spaces, Colloq. Math., 100 (2004), 221-242. MR 2107518 (2005i:35113)
  • 9. J.W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, 2000. MR 1778284 (2002f:37132)
  • 10. J.W. Cholewa and T. Dlotko, Hyperbolic equations in uniform spaces, Bulletin of The Polish Academy of Sciences Mathematics, 52 (2004), 249-263. MR 2127062 (2006b:35225)
  • 11. J.W. Cholewa and T. Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Mathematical Journal, 54 (2004), 991-1013. MR 2099352 (2005g:35148)
  • 12. J.W. Cholewa and T. Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal. TMA, 64 (2006), 174-187. MR 2183836 (2006f:35183)
  • 13. M. Conti, V. Pata and M. Squassina, Strongly damped wave equations on $ \mathbb{R}^3$ with critical nonlinearities, Communications in Applied Analysis, 9 (2005), 161-176. MR 2168756 (2006d:35165)
  • 14. M.A. Efendiev, A. Miranville and S.V. Zelik, Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation, Proc. R. Soc. Lond. A, 460 (2004), 1107-1129. MR 2133858 (2005m:37190)
  • 15. M.A. Efendiev and S.V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688. MR 1815444 (2001m:35035)
  • 16. E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations in $ \mathbb{R}^N$, Differential Integral Equations, 9 (1996), 1147-1156. MR 1392099 (97f:35138)
  • 17. P. Fabrie, C. Galushinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singular perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10 (2004), 211-238. MR 2026192 (2006c:37088)
  • 18. E. Feireisl, Ph. Laurencot, F. Simondon and H. Toure, Compact attractors for reaction diffusion equations in $ \mathbb{R}^N$, C. R. Acad. Sci. Paris. Sér. I. Math., 319 (1994), 147-151. MR 1288394 (95f:35110)
  • 19. E. Feireisl, Ph. Laurencot and F. Simondon, Global attractors for degenerate parabolic equations on unbounded domains, J. Diff. Equations, 129 (1996), 239-261. MR 1404383 (97j:35082)
  • 20. C. Gatti, A. Miranville, V. Pata and S.V. Zelik, Attractors for semilinear equations of viscoelasticity with very low dissipation, R. Mountain J. Math., 38 (2008), 1117-1138.
  • 21. J.K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. MR 941371 (89g:58059)
  • 22. N. Karachalios and N. Stavrakakis, Existence of a global attractor for semilinear dissipative wave equations on $ \mathbb{R}^N$, J. Diff. Equations, 157 (1999), 183-205. MR 1710020 (2000e:35151)
  • 23. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal., 58 (1975), 181-205. MR 0390516 (52:11341)
  • 24. O.A. Ladyzhenskaya, Attractors for semigroups and evolution equations, Leizioni Lincei, Cambridge Univ. Press, Cambridge, New York, 1991. MR 1133627 (92k:58040)
  • 25. A. Mielke, The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity, 10 (1997), 199-222. MR 1430749 (97m:35242)
  • 26. A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity, 8 (1995) 743-768. MR 1355041 (97e:58207)
  • 27. V. Pata and M. Squassina, On the strongly damped wave equation, Comm. Math. Phys., 253 (2005), 511-533. MR 2116726 (2005k:35291)
  • 28. V. Pata and S.V. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506. MR 2229785 (2007f:35257)
  • 29. C. Sun, D. Cao and J. Duan, Non-autonomous wave dynamics with memory -- Asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst., Ser. B, 9 (2008), 743-761. MR 2379435
  • 30. C. Sun and M. Yang, Attractors of strongly damped wave equations: asymptotic regularity and exponential attraction, submitted.
  • 31. C. Sun, M. Yang and C. Zhong, Global attractors for hyperbolic equations with critical exponent in locally uniform spaces, submitted.
  • 32. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. MR 1441312 (98b:58056)
  • 33. M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces II: Infinite-dimensional exponential attractors and their approximation, preparation.
  • 34. S.V. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dyn. Syst., 7 (2001), 593-641. MR 1815770 (2001m:35230)
  • 35. S.V. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Comm. Pure Appl. Anal., 3 (2004), 921-934. MR 2106304 (2005h:35249)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37L05, 35B40, 35B41

Retrieve articles in all journals with MSC (2000): 37L05, 35B40, 35B41

Additional Information

Meihua Yang
Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China – and – Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

Chunyou Sun
Affiliation: Department of Mathematics, Lanzhou University, Lanzhou, 730000, People’s Republic of China – and – Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, 100080, People’s Republic of China

Keywords: Strongly damped wave equation, locally uniform spaces, critical exponent, asymptotic regularity, attractors.
Received by editor(s): May 18, 2007
Published electronically: September 29, 2008
Additional Notes: This work was supported by the NSFC Grants 10601021 and 10726024 and the China Postdoctoral Science Foundation.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society