# American Institute of Mathematical Sciences

August  2015, 35(8): 3799-3825. doi: 10.3934/dcds.2015.35.3799

## Global attractor for weakly damped gKdV equations in higher sobolev spaces

 1 School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China

Received  August 2014 Revised  December 2014 Published  February 2015

Long time behavior of solutions for weakly damped gKdV equations on the real line is studied. With some weak regularity assumptions on the force $f$, we prove the existence of global attractor in $H^s$ for any $s\geq 1$. The asymptotic compactness of solution semigroup is shown by Ball's energy method and Goubet's high-low frequency decomposition if $s$ is an integer and not an integer, respectively.
Citation: Ming Wang. Global attractor for weakly damped gKdV equations in higher sobolev spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3799-3825. doi: 10.3934/dcds.2015.35.3799
##### References:
 [1] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, International Mathematics Research Notices, 6 (1996), 277-304. doi: 10.1155/S1073792896000207. [2] I. Chueshov and I. Lasiecka, Long-time behaviour of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. [3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}^{N}$ and $T$, Journal of the American Mathematical Society, 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. [4] O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete and Continuous Dynamical Systems, 6 (2000), 625-644. [5] O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, Journal of Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163. [6] A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential and Integral Equations, 18 (2005), 1333-1339. [7] J. K. Hale, Asmptotic Behavior of Dissipative Systems, Providence, RI: American Mathematical Society, 1988. [8] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, Journal of the American Mathematical Society, 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0. [9] C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3. [10] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized korteweg-de vries equation via the contraction principle, Communications on Pure and Applied Mathematics, 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. [11] H. Koch and J. Marzuola, Small data scattering and soliton stability in $\dotH^{-\frac{1}{6}}$ for the quartic KdV equation, Analysis & PDE, 5 (2012), 145-198. doi: 10.2140/apde.2012.5.145. [12] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave, Philos. Mag., 39 (1895), 422-443. [13] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer, New York, 2009. [14] K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, Journal of Differential Equations, 170 (2001), 281-316. doi: 10.1006/jdeq.2000.3827. [15] Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. [16] Y. Martel and F. Merle, Blow up in finite time and dynamics of blow up solutions for the $L^2$-critical generalized KdV equation, Journal of the American Mathematical Society, 15 (2002), 617-664. [17] Y. Martel and F. Merle, Review of long time asymptotics and collision of solitons for the quartic generalized Korteweg-de Vries equation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 141 (2011), 287-317. doi: 10.1017/S030821051000003X. [18] Y. Martel, F. Merle and P. Raphaël, Blow up for the critical generalized Korteweg-de Vries equation. I: Dynamics near the soliton, Acta Mathematica, 212 (2014), 59-140. doi: 10.1007/s11511-014-0109-2. [19] Y. Martel, F. Merle and T. P. Tsai, Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations, Communications in mathematical physics, 231 (2002), 347-373. [20] I. Moise and R. Rosa, On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation, Advances in Differential Equations, 2 (1997), 257-296. [21] L. Molinet, Global attractor and asymptotic smoothing effects for the weakly damped cubic Schrödinger equation in $L^ 2 (\mathbbT)$, Dynamics of Partial Differential Equations, 6 (2009), 15-34. doi: 10.4310/DPDE.2009.v6.n1.a2. [22] R. Rosa, The global attractor of a weakly damped, forced Korteweg-de Vries equation in $H^1(\mathbbR)$, VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999). Mat. Contemp., 19 (2000), 129-152. [23] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [24] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbb{R}$, Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399. [25] G. Staffilani, On the growth of high Sobolev norms of solutions for $KdV$ and Schrödinger equations, Duke Mathematical Journal, 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X. [26] C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59 (2008), 51-81. doi: 10.3233/ASY-2008-0886. [27] C. Sun, L. Yang and J. Duan, Asymptotic behavior for a semilinear second order evolution equation, Transactions of the American Mathematical Society, 363 (2011), 6085-6109. doi: 10.1090/S0002-9947-2011-05373-0. [28] C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion with distribution derivatives in unbounded domains, Nonlinear Anal.: Theory, Methods Appl., 63 (2005), 49-65. doi: 10.1016/j.na.2005.04.034. [29] T. Tao, Nonlinear dispersive equations: Local and global analysis, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. [30] T. Tao, Scattering for the quartic generalised Korteweg-de Vries equation, Journal of Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019. [31] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [32] B. Wang, Regularity of attractors for the Benjamin-Bona-Mahony equation, Journal of Physics A: Mathematical and General, 31 (1998), 7635-7645. doi: 10.1088/0305-4470/31/37/021. [33] M. Wang, D. Li, C. Zhang and Y. Tang, Long time behavior of gKdV equations, Journal of Mathematical Analysis and Applications, 390 (2012), 136-150. doi: 10.1016/j.jmaa.2012.01.031. [34] Y. Wu, The Cauchy problem of the Schrodinger-Korteweg-De Vries system, Differential And Integral Equations, 23 (2010), 569-600. [35] M. Yang and C. Sun, Exponential attractors for the strongly damped wave equations, Nonlinear Analysis: Real World Applications, 11 (2010), 913-919. doi: 10.1016/j.nonrwa.2009.01.022. [36] Y. Xie, Q. Li, C. Huang et al., Attractors for the semilinear reaction-diffusion equation with distribution derivatives, Journal of Mathematical Physics, 54 (2013), 092701, 11pp. doi: 10.1063/1.4818983.

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##### References:
 [1] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, International Mathematics Research Notices, 6 (1996), 277-304. doi: 10.1155/S1073792896000207. [2] I. Chueshov and I. Lasiecka, Long-time behaviour of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. [3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}^{N}$ and $T$, Journal of the American Mathematical Society, 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. [4] O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete and Continuous Dynamical Systems, 6 (2000), 625-644. [5] O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line, Journal of Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163. [6] A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential and Integral Equations, 18 (2005), 1333-1339. [7] J. K. Hale, Asmptotic Behavior of Dissipative Systems, Providence, RI: American Mathematical Society, 1988. [8] C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, Journal of the American Mathematical Society, 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0. [9] C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3. [10] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized korteweg-de vries equation via the contraction principle, Communications on Pure and Applied Mathematics, 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. [11] H. Koch and J. Marzuola, Small data scattering and soliton stability in $\dotH^{-\frac{1}{6}}$ for the quartic KdV equation, Analysis & PDE, 5 (2012), 145-198. doi: 10.2140/apde.2012.5.145. [12] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave, Philos. Mag., 39 (1895), 422-443. [13] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer, New York, 2009. [14] K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, Journal of Differential Equations, 170 (2001), 281-316. doi: 10.1006/jdeq.2000.3827. [15] Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. [16] Y. Martel and F. Merle, Blow up in finite time and dynamics of blow up solutions for the $L^2$-critical generalized KdV equation, Journal of the American Mathematical Society, 15 (2002), 617-664. [17] Y. Martel and F. Merle, Review of long time asymptotics and collision of solitons for the quartic generalized Korteweg-de Vries equation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 141 (2011), 287-317. doi: 10.1017/S030821051000003X. [18] Y. Martel, F. Merle and P. Raphaël, Blow up for the critical generalized Korteweg-de Vries equation. I: Dynamics near the soliton, Acta Mathematica, 212 (2014), 59-140. doi: 10.1007/s11511-014-0109-2. [19] Y. Martel, F. Merle and T. P. Tsai, Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations, Communications in mathematical physics, 231 (2002), 347-373. [20] I. Moise and R. Rosa, On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation, Advances in Differential Equations, 2 (1997), 257-296. [21] L. Molinet, Global attractor and asymptotic smoothing effects for the weakly damped cubic Schrödinger equation in $L^ 2 (\mathbbT)$, Dynamics of Partial Differential Equations, 6 (2009), 15-34. doi: 10.4310/DPDE.2009.v6.n1.a2. [22] R. Rosa, The global attractor of a weakly damped, forced Korteweg-de Vries equation in $H^1(\mathbbR)$, VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999). Mat. Contemp., 19 (2000), 129-152. [23] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9. [24] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbb{R}$, Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399. [25] G. Staffilani, On the growth of high Sobolev norms of solutions for $KdV$ and Schrödinger equations, Duke Mathematical Journal, 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X. [26] C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59 (2008), 51-81. doi: 10.3233/ASY-2008-0886. [27] C. Sun, L. Yang and J. Duan, Asymptotic behavior for a semilinear second order evolution equation, Transactions of the American Mathematical Society, 363 (2011), 6085-6109. doi: 10.1090/S0002-9947-2011-05373-0. [28] C. Sun and C. Zhong, Attractors for the semilinear reaction-diffusion with distribution derivatives in unbounded domains, Nonlinear Anal.: Theory, Methods Appl., 63 (2005), 49-65. doi: 10.1016/j.na.2005.04.034. [29] T. Tao, Nonlinear dispersive equations: Local and global analysis, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. [30] T. Tao, Scattering for the quartic generalised Korteweg-de Vries equation, Journal of Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019. [31] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [32] B. Wang, Regularity of attractors for the Benjamin-Bona-Mahony equation, Journal of Physics A: Mathematical and General, 31 (1998), 7635-7645. doi: 10.1088/0305-4470/31/37/021. [33] M. Wang, D. Li, C. Zhang and Y. Tang, Long time behavior of gKdV equations, Journal of Mathematical Analysis and Applications, 390 (2012), 136-150. doi: 10.1016/j.jmaa.2012.01.031. [34] Y. Wu, The Cauchy problem of the Schrodinger-Korteweg-De Vries system, Differential And Integral Equations, 23 (2010), 569-600. [35] M. Yang and C. Sun, Exponential attractors for the strongly damped wave equations, Nonlinear Analysis: Real World Applications, 11 (2010), 913-919. doi: 10.1016/j.nonrwa.2009.01.022. [36] Y. Xie, Q. Li, C. Huang et al., Attractors for the semilinear reaction-diffusion equation with distribution derivatives, Journal of Mathematical Physics, 54 (2013), 092701, 11pp. doi: 10.1063/1.4818983.
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