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

Yang, Meihua; Sun, Chunyou

doi:10.1090/S0002-9947-08-04680-1

Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity
HTML articles powered by AMS MathViewer

by Meihua Yang and Chunyou Sun PDF

Trans. Amer. Math. Soc. 361 (2009), 1069-1101 Request permission

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 op- timal 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

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
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, DOI 10.1142/S0218202504003234
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, DOI 10.1016/S0168-2024(08)70270-4
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, DOI 10.1017/S0308210500031498
Veronica Belleri and Vittorino Pata, Attractors for semilinear strongly damped wave equations on $\Bbb R^3$, Discrete Contin. Dynam. Systems 7 (2001), no. 4, 719–735. MR 1849655, DOI 10.3934/dcds.2001.7.719
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, DOI 10.2140/pjm.2002.207.287
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, DOI 10.1017/S0004972700040296
Alexandre N. Carvalho and Tomasz Dlotko, Partly dissipative systems in uniformly local spaces, Colloq. Math. 100 (2004), no. 2, 221–242. MR 2107518, DOI 10.4064/cm100-2-6
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, DOI 10.1017/CBO9780511526404
J. W. Cholewa and Tomasz Dlotko, Hyperbolic equations in uniform spaces, Bull. Pol. Acad. Sci. Math. 52 (2004), no. 3, 249–263. MR 2127062, DOI 10.4064/ba52-3-5
Jan W. Cholewa and Tomasz Dlotko, Cauchy problems in weighted Lebesgue spaces, Czechoslovak Math. J. 54(129) (2004), no. 4, 991–1013. MR 2099352, DOI 10.1007/s10587-004-6447-z
J. W. Cholewa and Tomasz Dlotko, Strongly damped wave equation in uniform spaces, Nonlinear Anal. 64 (2006), no. 1, 174–187. MR 2183836, DOI 10.1016/j.na.2005.06.021
Monica Conti, Vittorino Pata, and Marco Squassina, Strongly damped wave equations on $\Bbb R^3$ with critical nonlinearities, Commun. Appl. Anal. 9 (2005), no. 2, 161–176. MR 2168756
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, DOI 10.1098/rspa.2003.1182
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, DOI 10.1002/cpa.1011
Eduard Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on $\mathbf R^N$, Differential Integral Equations 9 (1996), no. 5, 1147–1156. MR 1392099
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, DOI 10.3934/dcds.2004.10.211
Eduard Feireisl, Philippe Laurençot, Frédérique Simondon, and Hamidou Touré, Compact attractors for reaction-diffusion equations in $\textbf {R}^N$, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 2, 147–151 (English, with English and French summaries). MR 1288394
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, DOI 10.1006/jdeq.1996.0117
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.
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
Nikos I. Karachalios and Nikos M. Stavrakakis, Existence of a global attractor for semilinear dissipative wave equations on $\textbf {R}^N$, J. Differential Equations 157 (1999), no. 1, 183–205. MR 1710020, DOI 10.1006/jdeq.1999.3618
Tosio Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58 (1975), no. 3, 181–205. MR 390516, DOI 10.1007/BF00280740
Olga Ladyzhenskaya, Attractors for semigroups and evolution equations, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1991. MR 1133627, DOI 10.1017/CBO9780511569418
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, DOI 10.1088/0951-7715/10/1/014
Alexander Mielke and Guido Schneider, Attractors for modulation equations on unbounded domains—existence and comparison, Nonlinearity 8 (1995), no. 5, 743–768. MR 1355041, DOI 10.1088/0951-7715/8/5/006
Vittorino Pata and Marco Squassina, On the strongly damped wave equation, Comm. Math. Phys. 253 (2005), no. 3, 511–533. MR 2116726, DOI 10.1007/s00220-004-1233-1
Vittorino Pata and Sergey Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity 19 (2006), no. 7, 1495–1506. MR 2229785, DOI 10.1088/0951-7715/19/7/001
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, DOI 10.3934/dcdsb.2008.9.743
C. Sun and M. Yang, Attractors of strongly damped wave equations: asymptotic regularity and exponential attraction, submitted.
C. Sun, M. Yang and C. Zhong, Global attractors for hyperbolic equations with critical exponent in locally uniform spaces, submitted.
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
M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces II: Infinite-dimensional exponential attractors and their approximation, preparation.
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, DOI 10.3934/dcds.2001.7.593
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, DOI 10.3934/cpaa.2004.3.921