# American Institute of Mathematical Sciences

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

## Time-dependent attractor for the Oscillon equation

 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.

Citation: Francesco Di Plinio, Gregory S. Duane, Roger Temam. Time-dependent attractor for the Oscillon equation. Discrete and 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. [2] L. Arnold, "Random Dynamical Systems," Springer-Verlag, Berlin, 1998. [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. [4] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $mathbb{R}^{3}$, Discrete Cont. Dynam. Syst., 7 (2001), 719-735. doi: doi:10.3934/dcds.2001.7.719. [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. [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. [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. [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. [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. [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. [11] V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. [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 [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. [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: doi:10.1007/BF02219225. [15] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: doi:10.1007/BF01193705. [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. [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. [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. [19] J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319. [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. [21] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [22] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,'' Recherches en Mathmatiques Appliques [Research in Applied Mathematics], 17, Masson, Paris, 1991. [23] O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'' Cambridge University Press, Cambridge, UK, 1991. [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. [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. [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. [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. [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. [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. [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. [31] M. Schroeder, "Fractals, Chaos, Power Laws,'' W. H. Freeman and Company, New York, 1991. [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. [33] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. [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. [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.

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. [2] L. Arnold, "Random Dynamical Systems," Springer-Verlag, Berlin, 1998. [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. [4] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equations on $mathbb{R}^{3}$, Discrete Cont. Dynam. Syst., 7 (2001), 719-735. doi: doi:10.3934/dcds.2001.7.719. [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. [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. [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. [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. [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. [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. [11] V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. [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 [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. [14] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. doi: doi:10.1007/BF02219225. [15] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: doi:10.1007/BF01193705. [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. [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. [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. [19] J.-M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66 (1987), 273-319. [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. [21] J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Mathematical Surveys and Monographs, 25, American Mathematical Society, Providence, RI, 1988. [22] A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications,'' Recherches en Mathmatiques Appliques [Research in Applied Mathematics], 17, Masson, Paris, 1991. [23] O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,'' Cambridge University Press, Cambridge, UK, 1991. [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. [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. [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. [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. [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. [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. [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. [31] M. Schroeder, "Fractals, Chaos, Power Laws,'' W. H. Freeman and Company, New York, 1991. [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. [33] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,'' 2nd edition Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. [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. [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.
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