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January  2011, 29(1): 141-167. doi: 10.3934/dcds.2011.29.141

Time-dependent attractor for the Oscillon equation

Francesco Di Plinio 1, , Gregory S. Duane 2, and  Roger Temam 3,

1. 

Indiana University Mathematics Department, Bloomington, IN 47405, United States
2. 

Rosenstiel School of Marine and Atmospheric Sciences, University of Miami, Miami, FL 33149, United States
3. 

The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405

Received  January 2010 Revised  May 2010 Published  September 2010

We investigate the asymptotic behavior of the nonautonomous evolution problem generated by the Oscillon equation

∂ tt $u(x,t) +H $ ∂ t$ u(x,t) -\e^{-2Ht}$ ∂ xx $ u(x,t) + V'(u(x,t)) =0, \quad (x,t)\in (0,1) \times \R,$

with periodic boundary conditions, where $H>0$ is the Hubble constant and $V$ is a nonlinear potential of arbitrary polynomial growth. After constructing a suitable dynamical framework to deal with the explicit time dependence of the energy of the solution, we establish the existence of a regular global attractor $\A=\A(t)$. The kernel sections $\A(t)$ have finite fractal dimension.

Keywords: nonautonomous attractors, Oscillon equation, fractal dimension..
Mathematics Subject Classification: Primary: 37L30, 35B41; Secondary: 83D0.
Citation: Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 141-167. doi: 10.3934/dcds.2011.29.141
References:
[1]

A. B. Adib, M. Gleiser and C. A. S. Almeida, Long lived oscillons from asymmetric bubbles: Existence and stability, Phys. Rev. D, 66 (2002), 085011. doi: doi:10.1103/PhysRevD.66.085011.  Google Scholar

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V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\R^3$, Discrete Cont. Dynam. Syst., 7 (2001), 719-735. doi: doi:10.3934/dcds.2001.7.719.  Google Scholar

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Z. Brzeźniak, M. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Related Fields, 95 (1993), 87-102. doi: doi:10.1007/BF01197339.  Google Scholar

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T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems, Dynam. Contin. Discrete Impuls. Systems A, 10 (2003), 491-514.  Google Scholar

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T. Caraballo, J. A. Langa and J. Valero, The dimension of attractors of nonautonomous partial differential equations, ANZIAM J., 45 (2003), 207-222. doi: doi:10.1017/S1446181100013274.  Google Scholar

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T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: doi:10.1016/j.na.2005.03.111.  Google Scholar

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T. Caraballo, G. Ł ukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 263-268.  Google Scholar

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D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dynam. Systems Theory, 2 (2002), 9-28.  Google Scholar

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V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.  Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, "Attractors of Equations of Mathematical Physics,'' American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002  Google Scholar

[13]

E. J. Copeland, M. Gleiser and H. R. Muller, Oscillons: Resonant configurations during bubble collapse, Phys. Rev. D., 52 (1995), 1920-1933. doi: doi:10.1103/PhysRevD.52.1920.  Google Scholar

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: doi:10.1007/BF02219225.  Google Scholar

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: doi:10.1007/BF01193705.  Google Scholar

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P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 221-238.  Google Scholar

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E. Farhi, N. Graham, V. Khemani, et al., An oscillon in the $SU(2)$ gauged Higgs model, Phys. Rev. D, 72 (2005), 101701. doi: doi:10.1103/PhysRevD.72.101701.  Google Scholar

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J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319.  Google Scholar

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M. Gleiser and A. Sornberger, Longlived localized field configurations in small lattices: Application to oscillons, Phys. Rev. E, 62 (2000), 1368-1374. doi: doi:10.1103/PhysRevE.62.1368.  Google Scholar

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J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

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A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,'' Recherches en Mathmatiques Appliques [Research in Applied Mathematics], 17, Masson, Paris, 1991.  Google Scholar

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O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'' Cambridge University Press, Cambridge, UK, 1991.  Google Scholar

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B. B. Mandelbrot, "The Fractal Geometry of Nature,'' Schriftenreihe fr den Referenten. [Series for the Referee] W. H. Freeman and Co., San Francisco, Calif., 1982.  Google Scholar

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A. Miranville and S. Zelik, "Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,'' Handbook of Differential Equations: Evolutionary Equations, Vol. IV, 103-200, Elsevier/North-Holland, Amsterdam, 2008.  Google Scholar

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P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: doi:10.1016/j.na.2009.02.065.  Google Scholar

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V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Comm. Pure Appl. Anal., 2 (2007), 481-486.  Google Scholar

[29]

A. Riotto, Are oscillons present during a first order electroweak phase transition?, Phys. Lett. B, 365 (1996), 64-71. doi: doi:10.1016/0370-2693(95)01239-7.  Google Scholar

[30]

B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics and Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), Dresden, 73 (1992), 185-192. Google Scholar

[31]

M. Schroeder, "Fractals, Chaos, Power Laws,'' W. H. Freeman and Company, New York, 1991.  Google Scholar

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C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665. doi: doi:10.1088/0951-7715/19/11/008.  Google Scholar

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R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar

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P. B. Umbanhower, F. Melo and H. L. Swinney, Localized excitations in a vertically vibrated granular layer, Nature, 382 (1996), 793-796. doi: doi:10.1038/382793a0.  Google Scholar

[35]

Y. Wang, Pullback attractors for nonautonomous wave equations with critical exponent, Nonlinear Anal., 68 (2008), 365-376. doi: doi:10.1016/j.na.2006.11.002.  Google Scholar

show all references

References:
[1]

A. B. Adib, M. Gleiser and C. A. S. Almeida, Long lived oscillons from asymmetric bubbles: Existence and stability, Phys. Rev. D, 66 (2002), 085011. doi: doi:10.1103/PhysRevD.66.085011.  Google Scholar

[2]

L. Arnold, "Random Dynamical Systems," Springer-Verlag, Berlin, 1998.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.  Google Scholar

[4]

V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $\R^3$, Discrete Cont. Dynam. Syst., 7 (2001), 719-735. doi: doi:10.3934/dcds.2001.7.719.  Google Scholar

[5]

Z. Brzeźniak, M. Capiński and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Probab. Theory Related Fields, 95 (1993), 87-102. doi: doi:10.1007/BF01197339.  Google Scholar

[6]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for nonautonomous and random dynamical systems, Dynam. Contin. Discrete Impuls. Systems A, 10 (2003), 491-514.  Google Scholar

[7]

T. Caraballo, J. A. Langa and J. Valero, The dimension of attractors of nonautonomous partial differential equations, ANZIAM J., 45 (2003), 207-222. doi: doi:10.1017/S1446181100013274.  Google Scholar

[8]

T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498. doi: doi:10.1016/j.na.2005.03.111.  Google Scholar

[9]

T. Caraballo, G. Ł ukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 263-268.  Google Scholar

[10]

D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dynam. Systems Theory, 2 (2002), 9-28.  Google Scholar

[11]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.  Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, "Attractors of Equations of Mathematical Physics,'' American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002  Google Scholar

[13]

E. J. Copeland, M. Gleiser and H. R. Muller, Oscillons: Resonant configurations during bubble collapse, Phys. Rev. D., 52 (1995), 1920-1933. doi: doi:10.1103/PhysRevD.52.1920.  Google Scholar

[14]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: doi:10.1007/BF02219225.  Google Scholar

[15]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: doi:10.1007/BF01193705.  Google Scholar

[16]

P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation, Discrete Cont. Dyn. Systems, 10 (2004), 221-238.  Google Scholar

[17]

E. Farhi, N. Graham, V. Khemani, et al., An oscillon in the $SU(2)$ gauged Higgs model, Phys. Rev. D, 72 (2005), 101701. doi: doi:10.1103/PhysRevD.72.101701.  Google Scholar

[18]

S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors, Proc. Amer. Math. Soc., 134 (2006), 117-127. doi: doi:10.1090/S0002-9939-05-08340-1.  Google Scholar

[19]

J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319.  Google Scholar

[20]

M. Gleiser and A. Sornberger, Longlived localized field configurations in small lattices: Application to oscillons, Phys. Rev. E, 62 (2000), 1368-1374. doi: doi:10.1103/PhysRevE.62.1368.  Google Scholar

[21]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[22]

A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,'' Recherches en Mathmatiques Appliques [Research in Applied Mathematics], 17, Masson, Paris, 1991.  Google Scholar

[23]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'' Cambridge University Press, Cambridge, UK, 1991.  Google Scholar

[24]

B. B. Mandelbrot, "The Fractal Geometry of Nature,'' Schriftenreihe fr den Referenten. [Series for the Referee] W. H. Freeman and Co., San Francisco, Calif., 1982.  Google Scholar

[25]

A. Miranville and S. Zelik, "Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,'' Handbook of Differential Equations: Evolutionary Equations, Vol. IV, 103-200, Elsevier/North-Holland, Amsterdam, 2008.  Google Scholar

[26]

P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonlinear Anal., 71 (2009), 3956-3963. doi: doi:10.1016/j.na.2009.02.065.  Google Scholar

[27]

I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Cont. Dynam. Syst., 10 (2004), 473-496. doi: doi:10.3934/dcds.2004.10.473.  Google Scholar

[28]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Comm. Pure Appl. Anal., 2 (2007), 481-486.  Google Scholar

[29]

A. Riotto, Are oscillons present during a first order electroweak phase transition?, Phys. Lett. B, 365 (1996), 64-71. doi: doi:10.1016/0370-2693(95)01239-7.  Google Scholar

[30]

B. Schmalfuß, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics and Nonlinear Dynamics: Attractor Approximation and Global Behaviour (eds. V. Reitmann, T. Riedrich and N. Koksch), Dresden, 73 (1992), 185-192. Google Scholar

[31]

M. Schroeder, "Fractals, Chaos, Power Laws,'' W. H. Freeman and Company, New York, 1991.  Google Scholar

[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. doi: doi:10.1088/0951-7715/19/11/008.  Google Scholar

[33]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997.  Google Scholar

[34]

P. B. Umbanhower, F. Melo and H. L. Swinney, Localized excitations in a vertically vibrated granular layer, Nature, 382 (1996), 793-796. doi: doi:10.1038/382793a0.  Google Scholar

[35]

Y. Wang, Pullback attractors for nonautonomous wave equations with critical exponent, Nonlinear Anal., 68 (2008), 365-376. doi: doi:10.1016/j.na.2006.11.002.  Google Scholar

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