Global attractor for weakly damped gKdV equations in higher sobolev spaces

Ming Wang

doi:10.3934/dcds.2015.35.3799

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

Global attractor for weakly damped gKdV equations in higher sobolev spaces

Ming Wang ^1,

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

Received August 2014 Revised December 2014 Published February 2015

Abstract
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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.

Keywords: Bourgain space., Generalized KdV equation, global attractor.

Mathematics Subject Classification: Primary: 35Q53, 35B40; Secondary: 35B4.

Citation: Ming Wang. Global attractor for weakly damped gKdV equations in higher sobolev spaces. Discrete & 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. Google Scholar
[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. Google Scholar
[3]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$, Journal of the American Mathematical Society, 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar
[4]	O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete and Continuous Dynamical Systems, 6 (2000), 625-644. Google Scholar
[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. Google Scholar
[6]	A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential and Integral Equations, 18 (2005), 1333-1339. Google Scholar
[7]	J. K. Hale, Asmptotic Behavior of Dissipative Systems, Providence, RI: American Mathematical Society, 1988. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer, New York, 2009. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar
[24]	V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbbR$, Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399. Google Scholar
[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. Google Scholar
[26]	C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59 (2008), 51-81. doi: 10.3233/ASY-2008-0886. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	Y. Wu, The Cauchy problem of the Schrodinger-Korteweg-De Vries system, Differential And Integral Equations, 23 (2010), 569-600. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

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. Google Scholar
[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. Google Scholar
[3]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$, Journal of the American Mathematical Society, 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar
[4]	O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete and Continuous Dynamical Systems, 6 (2000), 625-644. Google Scholar
[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. Google Scholar
[6]	A. Grünrock, A bilinear Airy-estimate with application to gKdV-3, Differential and Integral Equations, 18 (2005), 1333-1339. Google Scholar
[7]	J. K. Hale, Asmptotic Behavior of Dissipative Systems, Providence, RI: American Mathematical Society, 1988. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Springer, New York, 2009. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar
[24]	V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on $\mathbbR$, Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399. Google Scholar
[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. Google Scholar
[26]	C. Sun and M. Yang, Dynamics of the nonclassical diffusion equations, Asymptotic Analysis, 59 (2008), 51-81. doi: 10.3233/ASY-2008-0886. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	Y. Wu, The Cauchy problem of the Schrodinger-Korteweg-De Vries system, Differential And Integral Equations, 23 (2010), 569-600. Google Scholar
[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. Google Scholar
[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. Google Scholar

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