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

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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 ) subset which attracts exponentially every initial -bounded set with respect to the -norm. Then, we show there is a -global attractor, which reflects the *strongly damped property* of to some extent.

**1.**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**, 10.1080/03605309208820866**2.**Jose M. Arrieta, Anibal Rodriguez-Bernal, Jan W. Cholewa, and Tomasz Dlotko,*Linear parabolic equations in locally uniform spaces*, Math. Models Methods Appl. Sci.**14**(2004), no. 2, 253–293. MR**2040897**, 10.1142/S0218202504003234**3.**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****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), no. 3-4, 221–243. MR**1084733**, 10.1017/S0308210500031498**5.**Veronica Belleri and Vittorino Pata,*Attractors for semilinear strongly damped wave equations on ℝ³*, Discrete Contin. Dynam. Systems**7**(2001), no. 4, 719–735. MR**1849655**, 10.3934/dcds.2001.7.719**6.**Alexandre N. Carvalho and Jan W. Cholewa,*Attractors for strongly damped wave equations with critical nonlinearities*, Pacific J. Math.**207**(2002), no. 2, 287–310. MR**1972247**, 10.2140/pjm.2002.207.287**7.**Alexandre N. Carvalho and Jan W. Cholewa,*Local well posedness for strongly damped wave equations with critical nonlinearities*, Bull. Austral. Math. Soc.**66**(2002), no. 3, 443–463. MR**1939206**, 10.1017/S0004972700040296**8.**Alexandre N. Carvalho and Tomasz Dlotko,*Partly dissipative systems in uniformly local spaces*, Colloq. Math.**100**(2004), no. 2, 221–242. MR**2107518**, 10.4064/cm100-2-6**9.**Jan W. Cholewa and Tomasz Dlotko,*Global attractors in abstract parabolic problems*, London Mathematical Society Lecture Note Series, vol. 278, Cambridge University Press, Cambridge, 2000. MR**1778284****10.**J. W. Cholewa and Tomasz Dlotko,*Hyperbolic equations in uniform spaces*, Bull. Pol. Acad. Sci. Math.**52**(2004), no. 3, 249–263. MR**2127062**, 10.4064/ba52-3-5**11.**Jan W. Cholewa and Tomasz Dlotko,*Cauchy problems in weighted Lebesgue spaces*, Czechoslovak Math. J.**54(129)**(2004), no. 4, 991–1013. MR**2099352**, 10.1007/s10587-004-6447-z**12.**J. W. Cholewa and Tomasz Dlotko,*Strongly damped wave equation in uniform spaces*, Nonlinear Anal.**64**(2006), no. 1, 174–187. MR**2183836**, 10.1016/j.na.2005.06.021**13.**Monica Conti, Vittorino Pata, and Marco Squassina,*Strongly damped wave equations on ℝ³ with critical nonlinearities*, Commun. Appl. Anal.**9**(2005), no. 2, 161–176. MR**2168756****14.**M. Efendiev, A. Miranville, and S. Zelik,*Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation*, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.**460**(2004), no. 2044, 1107–1129. MR**2133858**, 10.1098/rspa.2003.1182**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), no. 6, 625–688. MR**1815444**, 10.1002/cpa.1011**16.**Eduard Feireisl,*Bounded, locally compact global attractors for semilinear damped wave equations on 𝐑^{𝐍}*, Differential Integral Equations**9**(1996), no. 5, 1147–1156. MR**1392099****17.**Pierre Fabrie, Cedric Galusinski, Alain Miranville, and Sergey Zelik,*Uniform exponential attractors for a singularly perturbed damped wave equation*, Discrete Contin. Dyn. Syst.**10**(2004), no. 1-2, 211–238. Partial differential equations and applications. MR**2026192**, 10.3934/dcds.2004.10.211**18.**Eduard Feireisl, Philippe Laurençot, Frédérique Simondon, and Hamidou Touré,*Compact attractors for reaction-diffusion equations in 𝑅^{𝑁}*, C. R. Acad. Sci. Paris Sér. I Math.**319**(1994), no. 2, 147–151 (English, with English and French summaries). MR**1288394****19.**E. Feireisl, Ph. Laurençot, and F. Simondon,*Global attractors for degenerate parabolic equations on unbounded domains*, J. Differential Equations**129**(1996), no. 2, 239–261. MR**1404383**, 10.1006/jdeq.1996.0117**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.**Jack K. Hale,*Asymptotic behavior of dissipative systems*, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR**941371****22.**Nikos I. Karachalios and Nikos M. Stavrakakis,*Existence of a global attractor for semilinear dissipative wave equations on 𝑅^{𝑁}*, J. Differential Equations**157**(1999), no. 1, 183–205. MR**1710020**, 10.1006/jdeq.1999.3618**23.**Tosio Kato,*The Cauchy problem for quasi-linear symmetric hyperbolic systems*, Arch. Rational Mech. Anal.**58**(1975), no. 3, 181–205. MR**0390516****24.**Olga Ladyzhenskaya,*Attractors for semigroups and evolution equations*, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. MR**1133627****25.**Alexander Mielke,*The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors*, Nonlinearity**10**(1997), no. 1, 199–222. MR**1430749**, 10.1088/0951-7715/10/1/014**26.**Alexander Mielke and Guido Schneider,*Attractors for modulation equations on unbounded domains—existence and comparison*, Nonlinearity**8**(1995), no. 5, 743–768. MR**1355041****27.**Vittorino Pata and Marco Squassina,*On the strongly damped wave equation*, Comm. Math. Phys.**253**(2005), no. 3, 511–533. MR**2116726**, 10.1007/s00220-004-1233-1**28.**Vittorino Pata and Sergey Zelik,*Smooth attractors for strongly damped wave equations*, Nonlinearity**19**(2006), no. 7, 1495–1506. MR**2229785**, 10.1088/0951-7715/19/7/001**29.**Chunyou Sun, Daomin Cao, and Jinqiao Duan,*Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor*, Discrete Contin. Dyn. Syst. Ser. B**9**(2008), no. 3-4, 743–761. MR**2379435**, 10.3934/dcdsb.2008.9.743**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.**Roger Temam,*Infinite-dimensional dynamical systems in mechanics and physics*, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR**1441312****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. Dynam. Systems**7**(2001), no. 3, 593–641. MR**1815770**, 10.3934/dcds.2001.7.593**35.**Sergey Zelik,*Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent*, Commun. Pure Appl. Anal.**3**(2004), no. 4, 921–934. MR**2106304**, 10.3934/cpaa.2004.3.921

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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

Email:
yangmeih@gmail.com

**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

Email:
cysun@amss.ac.cn, sunchunyou@gmail.com

DOI:
https://doi.org/10.1090/S0002-9947-08-04680-1

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.