The Global Attractor of a Damped, Forced Hirota Equation in H 1

Guo, Boling ; Huo, Zhaohui

Canadian mathematical bulletin, Tome 53 (2010) no. 2, pp. 295-310

Résumé

The existence of the global attractor of a damped forced Hirota equation in the phase space ${{H}^{1}}\left( \mathbb{R} \right)$ is proved. The main idea is to establish the so-called asymptotic compactness property of the solution operator by energy equation approach.

Détail

DOI : 10.4153/CMB-2010-021-3

Mots-clés : 35Q53, 35B40, 35B41, 37L30, global attractor, Fourier restriction norm, damping system, asymptotic compactness

Bibliographie

[1] [1] Abergel, F., Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains. J. Differential Equations 83(1990), no. 1, 85–108. doi:10.1016/0022-0396(90)90070-6 Google Scholar

[2] [2] Bourgain, J., Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equation. Geom. Funct. Anal. 3(1993), no. 2, 107–156. doi:10.1007/BF01896020 Google Scholar

[3] [3] Bourgain, J., Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equation. II. The KdV equation. Geom. Funct. Anal. 3(1993), no. 3, 209–262. doi:10.1007/BF01895688 Google Scholar

[4] [4] Ghidaglia, J.-M., Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time. J. Differential Equations 74(1988), no. 2, 369–390. doi:10.1016/0022-0396(88)90010-1 Google Scholar

[5] [5] Ghidaglia, J.-M., A note on the strong convergence towards attractors of damped forced KdV equations. J. Differential Equations 110(1994), no. 2, 356–359. doi:10.1006/jdeq.1994.1071 Google Scholar

[6] [6] Goubet, O., Regularity of the global attractor for a weakly damped nonlinear Schrödinger equation in ℝ2 . Adv. Differential Equations 3(1998), no. 3, 337–360. Google Scholar

[7] [7] Goubet, O., Asymptotic smoothing effect for weakly damped Korteweg-de Vries equations. Discrete Contin. Dynam. Systems, 6(2000), no. 3, 625–644. Google Scholar

[8] [8] Goubet, O. and Rosa, R., Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line. J. Differential Equations 185(2002), no. 1, 25–53. doi:10.1006/jdeq.2001.4163 Google Scholar

[9] [9] Guo, B. and Tan, S., Global smooth solution for nonlinear evolution equation of Hirota type. Sci. China, Ser. A 35(1992), no. 12, 1425–1433. Google Scholar

[10] [10] Hasegawa, A., Optical Solitons in Fibers. Second edition. Springer, Berlin, 1990. Google Scholar

[11] [11] Hasegawa, A. and Kodama, Y., Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electronics. 23(1987), 510–524. Google Scholar

[12] [12] Huo, Z. and Guo, B., Well-posedness of the Cauchy problem for the Hirota equation in Sobolev spaces Hs . Nonlinear Anal. 60(2005), no. 6, 1093–1110. doi:10.1016/j.na.2004.10.011 Google Scholar

[13] [13] Huo, Z. and Jia, Y., Well-posedness for the Cauchy problem to the Hirota equation in Sobolev spaces of negative indices. Chinese Ann. Math. Ser. B 26(2005), no. 1, 75–88. doi:10.1142/S0252959905000075 Google Scholar

[14] [14] Karachalios, N. and Stavrakakis, N., Global attractor for the weakly damped driven Schrödinger equation in H2R . NoDEA Nonlinear Differential Equations Appl. 9(2002), no. 3, 347–360. doi:10.1007/s00030-002-8132-y Google Scholar

[15] [15] Kenig, C. E., Ponce, G., and Vega, L., The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71(1993), no. 1, 1–21. doi:10.1215/S0012-7094-93-07101-3 Google Scholar

[16] [16] Kenig, C. E., Ponce, G., and Vega, L., A bilinear estimate with applications to the Kdv equation. J. Amer. Math. Soc. 9(1996), no. 2, 573–603. doi:10.1090/S0894-0347-96-00200-7 Google Scholar

[17] [17] Kodama, Y., Optical solitons in a monomode fiber. J. Statist. Phys. 39(1985), no. 5–6, 597–614. doi:10.1007/BF01008354 Google Scholar

[18] [18] Ladyzhenskaya, O., Attractor for Semigroups and Evolution Equations. Lezioni Lincei, Cambridge University Press, 1991. Google Scholar

[19] [19] Laurey, C., The Cauchy problem for a third order nonlinear Schrödinger equation. Nonlinear Anal. 29(1997), no. 2, 121–158. doi:10.1016/S0362-546X(96)00081-8 Google Scholar

[20] [20] Moise, I. and Rosa, R., On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation. Adv. Differential Equations 2(1997), no. 2, 275–296. Google Scholar

[21] [21] Moise, I., Rosa, R., and Wang, X., Attractors for non-compact semigroups via energy equations. Nonlinearity 11(1998), no. 5, 1369–1393. doi:10.1088/0951-7715/11/5/012 Google Scholar

[22] [22] Rosa, R., The global attractor of a weakly damped, forced Korteweg-de Vries equation in H 1(ℝ) . Mat. Contemp. 19(2000), 129–152. Google Scholar

[23] [23] Tan, S., Han, Y., Long time behavior of solutions to general nonlinear evolution equations. Chinese Ann. Math. Ser. A 16(1995), no. 2, 127–141. Google Scholar

[24] [24] Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics. Second edition. Applied Mathematical Sciences 68, Springer-Verlag, New York, 1997. Google Scholar