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Global attractor for weakly damped gKdV equations in higher sobolev spaces
1. | School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China |
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. |
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. |
[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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