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 & Continuous Dynamical Systems - A, 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. 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). 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. 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. 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. 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. Google Scholar

[7]

J. K. Hale, Asmptotic Behavior of Dissipative Systems,, Providence, (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. 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. 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. 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. 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. Google Scholar

[13]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Springer, (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. 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. 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. 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. 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. 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. 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. 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. 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, 19 (2000), 129. Google Scholar

[23]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer, (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. 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. 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. 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. 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, 63 (2005), 49. 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, (2006). Google Scholar

[30]

T. Tao, Scattering for the quartic generalised Korteweg-de Vries equation,, Journal of Differential Equations, 232 (2007), 623. doi: 10.1016/j.jde.2006.07.019. Google Scholar

[31]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed.,, Springer, (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. 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. 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. Google Scholar

[35]

M. Yang and C. Sun, Exponential attractors for the strongly damped wave equations,, Nonlinear Analysis: Real World Applications, 11 (2010), 913. 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). 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. 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). 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. 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. 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. 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. Google Scholar

[7]

J. K. Hale, Asmptotic Behavior of Dissipative Systems,, Providence, (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. 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. 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. 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. 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. Google Scholar

[13]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Springer, (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. 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. 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. 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. 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. 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. 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. 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. 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, 19 (2000), 129. Google Scholar

[23]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Springer, (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. 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. 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. 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. 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, 63 (2005), 49. 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, (2006). Google Scholar

[30]

T. Tao, Scattering for the quartic generalised Korteweg-de Vries equation,, Journal of Differential Equations, 232 (2007), 623. doi: 10.1016/j.jde.2006.07.019. Google Scholar

[31]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed.,, Springer, (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. 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. 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. Google Scholar

[35]

M. Yang and C. Sun, Exponential attractors for the strongly damped wave equations,, Nonlinear Analysis: Real World Applications, 11 (2010), 913. 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). doi: 10.1063/1.4818983. Google Scholar

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