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Volumn 10, Issue 1-2, 2004, Pages 31-52

Global attractors for damped semilinear wave equations

Author keywords

Asymptotically compact; Asymptotically smooth; Damped; Generalized semi flow; Global attractor; Kneser's property; Lyapunov function; Nonuniqueness; Semilinear wave equation; Weak solution; Weakly continuous

Indexed keywords


EID: 2142768721     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2004.10.31     Document Type: Article
Times cited : (429)

References (73)
  • 2
    • 0042887224 scopus 로고
    • Maximal attractors of semigroups corresponding to evolution differential equations
    • A.V. Babin and M.I. Vishik. Maximal attractors of semigroups corresponding to evolution differential equations. Mat. Sbornik, 126:397-419, 1985. English translation in Math. USSR Sbornik 54(1986) pp. 387-408.
    • (1985) Mat. Sbornik , vol.126 , pp. 397-419
    • Babin, A.V.1    Vishik, M.I.2
  • 3
    • 67649394024 scopus 로고
    • English translation
    • A.V. Babin and M.I. Vishik. Maximal attractors of semigroups corresponding to evolution differential equations. Mat. Sbornik, 126:397-419, 1985. English translation in Math. USSR Sbornik 54(1986) pp. 387-408.
    • (1986) Math. USSR Sbornik , vol.54 , pp. 387-408
  • 6
    • 84966262522 scopus 로고
    • Strongly continuous semigroups, weak solutions, and the variation of constants formula
    • J.M. Ball. Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Amer. Math. Soc., 64:370-373, 1977.
    • (1977) Proc. Amer. Math. Soc. , vol.64 , pp. 370-373
    • Ball, J.M.1
  • 7
    • 0000262385 scopus 로고
    • On the asymptotic behaviour of generalized processes, with applications to non-linear evolution equations
    • J.M. Ball. On the asymptotic behaviour of generalized processes, with applications to non-linear evolution equations. J. Differential Eqns, 27:224-265, 1978.
    • (1978) J. Differential Eqns , vol.27 , pp. 224-265
    • Ball, J.M.1
  • 8
    • 0000938708 scopus 로고    scopus 로고
    • Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations
    • J.M. Ball. Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations. Nonlinear Science, 7:475-502, 1997. Erratum, ibid 8:233,1998; corrected version appears in Mechanics: from Theory to Computation, Springer, 2000.
    • (1997) Nonlinear Science , vol.7 , pp. 475-502
    • Ball, J.M.1
  • 9
    • 0000938708 scopus 로고    scopus 로고
    • Erratum
    • J.M. Ball. Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations. Nonlinear Science, 7:475-502, 1997. Erratum, ibid 8:233,1998; corrected version appears in Mechanics: from Theory to Computation, Springer, 2000.
    • (1998) Nonlinear Science , vol.8 , pp. 233
  • 10
    • 0000938708 scopus 로고    scopus 로고
    • corrected version appears, Springer
    • J.M. Ball. Continuity properties and attractors of generalized semiflows and the Navier-Stokes equations. Nonlinear Science, 7:475-502, 1997. Erratum, ibid 8:233,1998; corrected version appears in Mechanics: from Theory to Computation, Springer, 2000.
    • (2000) Mechanics: From Theory to Computation
  • 11
    • 0021296252 scopus 로고
    • Quasiconvexity at the boundary, positivity of the second variation, and elastic stability
    • J.M. Ball and J.E. Marsden. Quasiconvexity at the boundary, positivity of the second variation, and elastic stability. Arch. Rat. Mech. Anal., 86:251-277, 1984.
    • (1984) Arch. Rat. Mech. Anal. , vol.86 , pp. 251-277
    • Ball, J.M.1    Marsden, J.E.2
  • 12
    • 0003222693 scopus 로고
    • On the theory of generalized dynamical systems
    • English translation by U.S. Department of Commerce, Office of Technical Services, Washington D.C. 20235
    • E.A. Barbashin. On the theory of generalized dynamical systems. Moskov. Gos. Ped. Inst. Učen. Zap., 2:110-133, 1948. English translation by U.S. Department of Commerce, Office of Technical Services, Washington D.C. 20235.
    • (1948) Moskov. Gos. Ped. Inst. Učen. Zap. , vol.2 , pp. 110-133
    • Barbashin, E.A.1
  • 13
    • 85070032222 scopus 로고    scopus 로고
    • Existence and dimension of the attractor of the Bénard problem on channel-like domains
    • To appear
    • M. Cabral, R. Rosa, and R. Temam. Existence and dimension of the attractor of the Bénard problem on channel-like domains. Discr. Contin. Dyn. Syst. To appear.
    • Discr. Contin. Dyn. Syst.
    • Cabral, M.1    Rosa, R.2    Temam, R.3
  • 16
    • 0039699639 scopus 로고    scopus 로고
    • Global attractor of nonlinear strain waves in elastic waveguides
    • Z. Dai and B.L. Guo. Global attractor of nonlinear strain waves in elastic waveguides. Acta Math. Sci. Ser. B Engl. Ed., 20:322-334, 2000.
    • (2000) Acta Math. Sci. Ser. B Engl. Ed. , vol.20 , pp. 322-334
    • Dai, Z.1    Guo, B.L.2
  • 18
    • 0030464283 scopus 로고    scopus 로고
    • Finite-dimensional attractors for a class of semilinear wave equations
    • A. Eden and V. Kalantarov. Finite-dimensional attractors for a class of semilinear wave equations. Turkish J. Math., 20:425-450, 1996.
    • (1996) Turkish J. Math. , vol.20 , pp. 425-450
    • Eden, A.1    Kalantarov, V.2
  • 19
    • 0000085549 scopus 로고
    • Finite-dimensional exponential attractors for semi-linear wave equations with damping
    • A. Eden, A.J. Milani, and B. Nicolaenko, Finite-dimensional exponential attractors for semi-linear wave equations with damping. J. Math. Anal. Appl., 169:408-419, 1992.
    • (1992) J. Math. Anal. Appl. , vol.169 , pp. 408-419
    • Eden, A.1    Milani, A.J.2    Nicolaenko, B.3
  • 20
    • 84973937932 scopus 로고
    • Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent
    • E. Feireisl. Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent. Proc. Roy. Soc. Edinburgh Sect. A, 125:1051-1062, 1995.
    • (1995) Proc. Roy. Soc. Edinburgh Sect. A , vol.125 , pp. 1051-1062
    • Feireisl, E.1
  • 21
    • 0000220290 scopus 로고
    • Global attractors for semilinear damped wave equations with supercritical exponent
    • E. Feireisl. Global attractors for semilinear damped wave equations with supercritical exponent. J. Differential Eqns, 116:431-447, 1995.
    • (1995) J. Differential Eqns , vol.116 , pp. 431-447
    • Feireisl, E.1
  • 22
    • 0004260327 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge, second edition, With additional material by M. Joshi
    • F.G. Friedlander. Introduction to the. theory of distributions. Cambridge University Press, Cambridge, second edition, 1998. With additional material by M. Joshi.
    • (1998) Introduction to the Theory of Distributions
    • Friedlander, F.G.1
  • 23
    • 24844467009 scopus 로고
    • Sur les injections entre espaces de Sobolev et espaces d'Orlicz et application au comportement à l'infini pour des équations des ondes semi-linéaires
    • T. Gallouët. Sur les injections entre espaces de Sobolev et espaces d'Orlicz et application au comportement à l'infini pour des équations des ondes semi-linéaires, Portugal. Math., 42:97-112, 1983/84.
    • (1983) Portugal. Math. , vol.42 , pp. 97-112
    • Gallouët, T.1
  • 24
    • 0001005263 scopus 로고
    • A note on the strong convergence towards attractors of damped forced KdV equations
    • J.-M. Ghidaglia. A note on the strong convergence towards attractors of damped forced KdV equations. J. Differential Eqns, 110:356-359, 1994.
    • (1994) J. Differential Eqns , vol.110 , pp. 356-359
    • Ghidaglia, J.-M.1
  • 25
    • 0034419781 scopus 로고    scopus 로고
    • Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations
    • O. Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete Contin. Dynam. Systems, 6:625-644, 2000.
    • (2000) Discrete Contin. Dynam. Systems , vol.6 , pp. 625-644
    • Goubet, O.1
  • 26
    • 0032017698 scopus 로고    scopus 로고
    • Attractor for dissipative Zakharov system
    • O. Goubet and I. Moise. Attractor for dissipative Zakharov system. Nonlinear Anal., 31:823-847, 1998.
    • (1998) Nonlinear Anal. , vol.31 , pp. 823-847
    • Goubet, O.1    Moise, I.2
  • 27
    • 0037058009 scopus 로고    scopus 로고
    • Asymptotic smoothing and the global attractor of a weakly damped, forced KdV equation on the real line
    • R. Goubet and R. Rosa. Asymptotic smoothing and the global attractor of a weakly damped, forced KdV equation on the real line. J. Differential Eqns, 185:25-53, 2002.
    • (2002) J. Differential Eqns , vol.185 , pp. 25-53
    • Goubet, R.1    Rosa, R.2
  • 30
    • 0002555899 scopus 로고
    • Convergence in gradient-like systems with applications to PDE
    • J.K. Hale and G. Raugel. Convergence in gradient-like systems with applications to PDE. Z. Angew. Math. Phys., 43:63-124, 1992.
    • (1992) Z. Angew. Math. Phys. , vol.43 , pp. 63-124
    • Hale, J.K.1    Raugel, G.2
  • 32
    • 0012120134 scopus 로고
    • Two remarks on dissipative hyperbolic problems
    • J-L Lions, editor
    • A. Haraux, Two remarks on dissipative hyperbolic problems. In J-L Lions, editor, Séminaires de Collège de France, 1984.
    • (1984) Séminaires de Collège de France
    • Haraux, A.1
  • 33
    • 0002796050 scopus 로고    scopus 로고
    • Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity
    • A. Haraux and M.A. Jendoubi. Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity. Calc. Var., 9:95-124, 1999.
    • (1999) Calc. Var. , vol.9 , pp. 95-124
    • Haraux, A.1    Jendoubi, M.A.2
  • 34
    • 0347786600 scopus 로고    scopus 로고
    • Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity
    • A. Haraux and M.A. Jendoubi. Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity. Asymptot. Anal., 26:21-36, 2001.
    • (2001) Asymptot. Anal. , vol.26 , pp. 21-36
    • Haraux, A.1    Jendoubi, M.A.2
  • 35
  • 36
    • 0034374595 scopus 로고    scopus 로고
    • On the dimension of the global attractor for a damped semilinear wave equation with critical exponent
    • Y. Huang, Z. Yi, and Z. Yin. On the dimension of the global attractor for a damped semilinear wave equation with critical exponent. J. Math. Phys., 41:4957-4966, 2000.
    • (2000) J. Math. Phys. , vol.41 , pp. 4957-4966
    • Huang, Y.1    Yi, Z.2    Yin, Z.3
  • 37
    • 0000898687 scopus 로고    scopus 로고
    • Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity
    • M.A. Jendoubi. Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity. J. Differential Eqns, 144:302-312, 1998.
    • (1998) J. Differential Eqns , vol.144 , pp. 302-312
    • Jendoubi, M.A.1
  • 39
    • 0035427421 scopus 로고    scopus 로고
    • Kneser families in infinite-dimensional spaces
    • Ser. A: Theory Methods
    • T. Kaminogo. Kneser families in infinite-dimensional spaces. Nonlinear Anal., 45(5, Ser. A: Theory Methods):613-627, 2001.
    • (2001) Nonlinear Anal. , vol.45 , Issue.5 , pp. 613-627
    • Kaminogo, T.1
  • 40
    • 0004743791 scopus 로고    scopus 로고
    • Kneser's property and mapping degree to multi-valued Poincaré map described by a semilinear parabolic partial differential equation
    • T. Kaminogo and N. Kikuchi. Kneser's property and mapping degree to multi-valued Poincaré map described by a semilinear parabolic partial differential equation. Nonlinear World, 4:381-390, 1997.
    • (1997) Nonlinear World , vol.4 , pp. 381-390
    • Kaminogo, T.1    Kikuchi, N.2
  • 41
    • 0037953627 scopus 로고
    • Minimal compact global attractor for a damped semilinear wave equation
    • L. Kapitanski. Minimal compact global attractor for a damped semilinear wave equation. Comm. Partial Differential Equations, 20:1303-1323, 1995.
    • (1995) Comm. Partial Differential Equations , vol.20 , pp. 1303-1323
    • Kapitanski, L.1
  • 42
    • 85070037668 scopus 로고    scopus 로고
    • Convergence towards attractors for a degenerate Ginzburg-Landau equation
    • To appear
    • N. Karachalios and N. Zographopoulos. Convergence towards attractors for a degenerate Ginzburg-Landau equation. Z. Angew. Math. Phys. To appear.
    • Z. Angew. Math. Phys.
    • Karachalios, N.1    Zographopoulos, N.2
  • 45
    • 38249004460 scopus 로고
    • Kneser's property for a parabolic partial differential equation
    • N. Kikuchi. Kneser's property for a parabolic partial differential equation. Nonlinear Anal., 20:205-213, 1993.
    • (1993) Nonlinear Anal. , vol.20 , pp. 205-213
    • Kikuchi, N.1
  • 46
    • 2142684404 scopus 로고
    • Kneser's property of solutions of nonlinear integral equations
    • N. Kikuchi and S. Nakagiri. Kneser's property of solutions of nonlinear integral equations. Funkcial. Ekvac., 17:57-66, 1974.
    • (1974) Funkcial. Ekvac. , vol.17 , pp. 57-66
    • Kikuchi, N.1    Nakagiri, S.2
  • 49
    • 84974178842 scopus 로고
    • Compact attractor for a nonlinear wave equation with critical exponent
    • O. Lopes. Compact attractor for a nonlinear wave equation with critical exponent. Proc. Roy. Soc. Edinburgh Sect. A, 115:61-64, 1990.
    • (1990) Proc. Roy. Soc. Edinburgh Sect. A , vol.115 , pp. 61-64
    • Lopes, O.1
  • 50
    • 0035285098 scopus 로고    scopus 로고
    • Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains
    • K. Lu and B. Wang. Global attractors for the Klein-Gordon- Schrödinger equation in unbounded domains. J. Differential Eqns, 170:281-316, 2001.
    • (2001) J. Differential Eqns , vol.170 , pp. 281-316
    • Lu, K.1    Wang, B.2
  • 51
    • 0002969389 scopus 로고    scopus 로고
    • On attractors of multivalued semi-flows and differential inclusions
    • V.S. Melnik and J. Valero. On attractors of multivalued semi-flows and differential inclusions. Set-Valued Anal., 6:83-111, 1998.
    • (1998) Set-valued Anal. , vol.6 , pp. 83-111
    • Melnik, V.S.1    Valero, J.2
  • 52
    • 0034562774 scopus 로고    scopus 로고
    • On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions
    • V.S. Melnik and J. Valero. On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions. Set-Valued Anal., 8:375-403, 2000.
    • (2000) Set-valued Anal. , vol.8 , pp. 375-403
    • Melnik, V.S.1    Valero, J.2
  • 53
    • 84894538982 scopus 로고    scopus 로고
    • On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation
    • I. Moise and R. Rosa. On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation. Adv. Differential Equations, 2:257-296, 1997.
    • (1997) Adv. Differential Equations , vol.2 , pp. 257-296
    • Moise, I.1    Rosa, R.2
  • 54
    • 85070027227 scopus 로고    scopus 로고
    • Existence of uniform attractors for noncompact nonautonomous dissipative systems via energy equations
    • To appear
    • I. Moise, R. Rosa, and X. Wang. Existence of uniform attractors for noncompact nonautonomous dissipative systems via energy equations. Discrete Contin. Dyn. Syst. To appear.
    • Discrete Contin. Dyn. Syst.
    • Moise, I.1    Rosa, R.2    Wang, X.3
  • 55
    • 0004263512 scopus 로고    scopus 로고
    • Attractors for non-compact semigroups via energy equations
    • I. Moise, R. Rosa, and X. Wang. Attractors for non-compact semigroups via energy equations. Nonlinearity, 11:1369-1393, 1998.
    • (1998) Nonlinearity , vol.11 , pp. 1369-1393
    • Moise, I.1    Rosa, R.2    Wang, X.3
  • 57
    • 24844473098 scopus 로고    scopus 로고
    • Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations
    • Proc. Equadiff 10
    • P. Poláčik. Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations. Math. Bohemica, pages 301-310, 2002. Proc. Equadiff 10.
    • (2002) Math. Bohemica , pp. 301-310
    • Poláčik, P.1
  • 58
    • 0038815248 scopus 로고    scopus 로고
    • Global attractors in partial differential equations
    • North-Holland, Amsterdam
    • G. Raugel. Global attractors in partial differential equations. In Handbook of dynamical systems, Vol. 2, pages 885-982. North-Holland, Amsterdam, 2002.
    • (2002) Handbook of Dynamical Systems , vol.2 , pp. 885-982
    • Raugel, G.1
  • 59
    • 0032014991 scopus 로고    scopus 로고
    • The global attractor for the 2D Navier-Stokes flow on some unbounded domains
    • R. Rosa. The global attractor for the 2D Navier-Stokes flow on some unbounded domains. Nonlinear Anal., 32:71-85, 1998.
    • (1998) Nonlinear Anal. , vol.32 , pp. 71-85
    • Rosa, R.1
  • 60
    • 0011620040 scopus 로고    scopus 로고
    • 1(ℝ)
    • R. Rosa. The global attractor of a weakly damped, forced Korteweg-de Vries equation in ff^R). Mat. Contemp., 19:129-152, 2000. VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999).
    • (2000) Mat. Contemp. , vol.19 , pp. 129-152
    • Rosa, R.1
  • 61
    • 85070033456 scopus 로고    scopus 로고
    • Rio de Janeiro
    • R. Rosa. The global attractor of a weakly damped, forced Korteweg-de Vries equation in ff^R). Mat. Contemp., 19:129-152, 2000. VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999).
    • (1999) VI Workshop on Partial Differential Equations, Part II
  • 62
    • 0002456303 scopus 로고    scopus 로고
    • Global attractors for the three-dimensional Navier-Stokes equations
    • G.R. Sell. Global attractors for the three-dimensional Navier-Stokes equations. J. Dynamics and Differential Eqns, 8:1-33, 1996.
    • (1996) J. Dynamics and Differential Eqns , vol.8 , pp. 1-33
    • Sell, G.R.1
  • 63
    • 84972553620 scopus 로고
    • Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
    • R.S. Strichartz. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 44:705-714, 1977.
    • (1977) Duke Math. J. , vol.44 , pp. 705-714
    • Strichartz, R.S.1
  • 65
    • 0000453612 scopus 로고
    • On imbeddings into Orlicz spaces and some applications
    • N.S. Trudinger. On imbeddings into Orlicz spaces and some applications. J. Math. Mech., 17:473-483, 1967.
    • (1967) J. Math. Mech. , vol.17 , pp. 473-483
    • Trudinger, N.S.1
  • 66
    • 0032544453 scopus 로고    scopus 로고
    • Regularity of attractors for the Benjamin-Bona-Mahony equation
    • B. Wang. Regularity of attractors for the Benjamin-Bona-Mahony equation. J. Phys. A, 31:7635-7645, 1998.
    • (1998) J. Phys. A , vol.31 , pp. 7635-7645
    • Wang, B.1
  • 67
    • 0039723135 scopus 로고    scopus 로고
    • Attractors for reaction-diffusion equations in unbounded domains
    • B. Wang. Attractors for reaction-diffusion equations in unbounded domains. Phys. D, 128:41-52, 1999.
    • (1999) Phys. D , vol.128 , pp. 41-52
    • Wang, B.1
  • 68
    • 0348195716 scopus 로고    scopus 로고
    • Attractors for the Klein-Gordon-Schrödinger equation
    • B. Wang and H. Lange. Attractors for the Klein-Gordon-Schrödinger equation. J. Math. Phys., 40:2445-2457, 1999.
    • (1999) J. Math. Phys. , vol.40 , pp. 2445-2457
    • Wang, B.1    Lange, H.2
  • 69
    • 0000796224 scopus 로고
    • An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors
    • X. Wang. An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors. Phys. D, 88:167-175, 1995.
    • (1995) Phys. D , vol.88 , pp. 167-175
    • Wang, X.1
  • 70
    • 0000479863 scopus 로고
    • A bifurcation problem for a nonlinear hyperbolic partial differential equation
    • G.F. Webb. A bifurcation problem for a nonlinear hyperbolic partial differential equation. SIAM J. Math. Anal., 10(5):922-932, 1979.
    • (1979) SIAM J. Math. Anal. , vol.10 , Issue.5 , pp. 922-932
    • Webb, G.F.1
  • 72
    • 2142780336 scopus 로고    scopus 로고
    • Dimension of the global attractor for damped nonlinear wave equations
    • S. Zhou. Dimension of the global attractor for damped nonlinear wave equations. Proc. Amer. Math. Soc., 12:3623-3631, 1999.
    • (1999) Proc. Amer. Math. Soc. , vol.12 , pp. 3623-3631
    • Zhou, S.1
  • 73
    • 0033096068 scopus 로고    scopus 로고
    • On dimension of the global attractor for damped nonlinear wave equations
    • S. Zhou. On dimension of the global attractor for damped nonlinear wave equations. J. Math. Phys., 40:1432-1438, 1999.
    • (1999) J. Math. Phys. , vol.40 , pp. 1432-1438
    • Zhou, S.1


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