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Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term
Qualitative analysis for a new generalized 2-component Camassa-Holm system
1. | College of Mathematics Science and National Center for Applied Mathematics, Chongqing Normal University, Chongqing 401331, China |
2. | College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China |
$ \begin{equation*} m_t = ( u m)_x+ u _xm- v m, \ \ n_t = ( u n)_x+ u _xn+ v n, \end{equation*} $ |
$ n+m = \frac{1}{2}( u _{xx}-4 u ) $ |
$ n-m = v _x $ |
$ H^s(\mathbb{R})\times H^{s-1}(\mathbb{R}) $ |
$ s>\frac{5}{2} $ |
References:
[1] |
A. Alsaedi, B. Ahmad, M. Kirane and B. T. Torebek,
Blowing-up solutions of the time-fractional dispersive equations, Adv. Nonlinear Anal., 10 (2021), 952-971.
doi: 10.1515/anona-2020-0153. |
[2] |
R. Beals and J. Szmigielski, A 2-Component Camassa-Holm Equation, Euler-Bernoulli Beam Problem and Non-Commutative Continued Fractions, arXiv: 2011.05964. |
[3] |
R. Camassa and D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[4] |
G. Chen, R. M. Chen and Y. Liu,
Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation, Indiana Univ. Math. J., 67 (2018), 2393-2433.
doi: 10.1512/iumj.2018.67.7510. |
[5] |
R. M. Chen, L. Fan, H. Gao and Y. Liu,
Breaking waves and solitary waves to the rotation-two-component Camassa-Holm system, SIAM J. Math. Anal., 49 (2017), 3573-3602.
doi: 10.1137/16M1073005. |
[6] |
R. M. Chen, W. Lian, D. Wang and R. Xu,
A rigidity property for the Novikov equation and the asymptotic stability of peakons, Arch. Ration. Mech. Anal., 241 (2021), 497-533.
doi: 10.1007/s00205-021-01658-z. |
[7] |
Y. Chen, H. Gao and Y. Liu,
On the Cauchy problem for the two-component Dullin-Gottwald-Holm system, Discrete Contin. Dyn. Syst., 33 (2013), 3407-3441.
doi: 10.3934/dcds.2013.33.3407. |
[8] |
J. Chu and J. Escher,
Variational formulations of steady rotational equatorial waves, Adv. Nonlinear Anal., 10 (2021), 534-547.
doi: 10.1515/anona-2020-0146. |
[9] |
G. M. Coclite and L. d. Ruvo,
Discontinuous solutions for the short-pulse master mode-locking equation, AIMS Math., 4 (2019), 437-462.
doi: 10.3934/math.2019.3.437. |
[10] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[11] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[12] |
A. Constantin and H. P. McKean,
A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.
doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. |
[13] |
A. Constantin and W. A. Strauss,
Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. |
[14] |
J. Escher and Z. Yin,
Well-posedness, blow-up phenomena, and global solutions for the b-equation, J. Reine Angew. Math., 624 (2008), 51-80.
doi: 10.1515/CRELLE.2008.080. |
[15] |
J. Escher, D. Henry, B. Kolev and T. Lyons,
Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271.
doi: 10.1007/s10231-014-0461-z. |
[16] |
E. D. Farnum and J. N. Kutz,
Dynamics of a low-dimensional model for short pulse mode locking, Photonics, 2 (2015), 865-882.
doi: 10.3390/photonics2030865. |
[17] |
A. S. Fokas and B. Fuchssteiner,
Symplectic structures, their B$\ddot{a}$klund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[18] |
G. L. Gui, Y. Liu and T. X. Tian,
Global existence and blow-up phenomena for the peakon b-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234.
doi: 10.1512/iumj.2008.57.3213. |
[19] |
A. Himonas and C. Holliman,
On well-posedness of the Degasperis-Procesi equation, Discrete Contin. Dyn. Syst., 31 (2011), 469-488.
doi: 10.3934/dcds.2011.31.469. |
[20] |
D. D. Holm and M. F. Staley,
Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.
doi: 10.1137/S1111111102410943. |
[21] |
R. I. Ivanov,
Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267-2280.
doi: 10.1098/rsta.2007.2007. |
[22] |
T. Kato,
Quasi-linear equations of evolution, with applications to partial differential equations in "Spectral theory and differential equations", Lecture Notes in Math., Springer Verlag, Berlin, 448 (1975), 25-70.
|
[23] |
T. Kato,
On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.
doi: 10.1007/BF01647967. |
[24] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[25] |
M. Li and Z. Yin, Global existence and local well-posedness of the single-cycle pulse equation, J. Math. Phys., 58 (2017), 101515, 16pp.
doi: 10.1063/1.5001381. |
[26] |
W. Luo and Z. Yin,
Gevrey regularity and analyticity for Camassa-Holm type systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1061-1079.
|
[27] |
Y. Liu, D. Pelinovsky and A. Sakovich,
Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291-310.
doi: 10.4310/DPDE.2009.v6.n4.a1. |
[28] |
Z. Luo, Z. Qiao and Z. Yin,
On the Cauchy problem for a modified Camassa-Holm equation, Monatsh. Math., 193 (2020), 857-877.
doi: 10.1007/s00605-020-01426-3. |
[29] |
Y. Mi, Y. Liu, B. Guo and T. Luo,
The Cauchy problem for a generalized Camassa-Holm equation, J. Differential Equations, 266 (2019), 6739-6770.
doi: 10.1016/j.jde.2018.11.019. |
[30] |
A. V. Mikhailov and V. S. Novikov,
Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.
doi: 10.1088/0305-4470/35/22/309. |
[31] |
B. Moon,
Single peaked traveling wave solutions to a generalized $\mu$-Novikov equation, Adv. Nonlinear Anal., 10 (2021), 66-75.
doi: 10.1515/anona-2020-0106. |
[32] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E(3), 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[33] |
D. Pelinovsky and A. Sakovich,
Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Comm. Partial Differential Equations, 35 (2010), 613-629.
doi: 10.1080/03605300903509104. |
[34] |
T. Schäfer and C. E. Wayne,
Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.
doi: 10.1016/j.physd.2004.04.007. |
[35] |
S. Wu, J. Escher and Z. Yin,
Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 633-645.
doi: 10.3934/dcdsb.2009.12.633. |
[36] |
R. Xu and Y. Yang,
Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Discrete Contin. Dyn. Syst., 40 (2020), 6507-6527.
doi: 10.3934/dcds.2020288. |
[37] |
K. Yan and Z. Yin,
Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269 (2011), 1113-1127.
doi: 10.1007/s00209-010-0775-5. |
[38] |
M. Yang, Y. Li and Z. Qiao,
Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system, Discrete Contin. Dyn. Syst., 40 (2020), 2475-2493.
doi: 10.3934/dcds.2020122. |
[39] |
Z. Zhaqilao, Q. Hu and Z. Qiao,
Multi-soliton solutions and the Cauchy problem for a two-component short pulse system, Nonlinearity, 30 (2017), 3773-3798.
doi: 10.1088/1361-6544/aa7e9c. |
[40] |
S. Zhou,
Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs, J. Evol. Equ., 14 (2014), 727-747.
doi: 10.1007/s00028-014-0236-4. |
[41] |
M. Zhu, Y. Liu and Y. Mi,
Wave-breaking phenomena and persistence properties for the nonlocal rotation-Camassa-Holm equation, Ann. Mat. Pura Appl., 199 (2020), 355-377.
doi: 10.1007/s10231-019-00882-5. |
show all references
References:
[1] |
A. Alsaedi, B. Ahmad, M. Kirane and B. T. Torebek,
Blowing-up solutions of the time-fractional dispersive equations, Adv. Nonlinear Anal., 10 (2021), 952-971.
doi: 10.1515/anona-2020-0153. |
[2] |
R. Beals and J. Szmigielski, A 2-Component Camassa-Holm Equation, Euler-Bernoulli Beam Problem and Non-Commutative Continued Fractions, arXiv: 2011.05964. |
[3] |
R. Camassa and D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[4] |
G. Chen, R. M. Chen and Y. Liu,
Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation, Indiana Univ. Math. J., 67 (2018), 2393-2433.
doi: 10.1512/iumj.2018.67.7510. |
[5] |
R. M. Chen, L. Fan, H. Gao and Y. Liu,
Breaking waves and solitary waves to the rotation-two-component Camassa-Holm system, SIAM J. Math. Anal., 49 (2017), 3573-3602.
doi: 10.1137/16M1073005. |
[6] |
R. M. Chen, W. Lian, D. Wang and R. Xu,
A rigidity property for the Novikov equation and the asymptotic stability of peakons, Arch. Ration. Mech. Anal., 241 (2021), 497-533.
doi: 10.1007/s00205-021-01658-z. |
[7] |
Y. Chen, H. Gao and Y. Liu,
On the Cauchy problem for the two-component Dullin-Gottwald-Holm system, Discrete Contin. Dyn. Syst., 33 (2013), 3407-3441.
doi: 10.3934/dcds.2013.33.3407. |
[8] |
J. Chu and J. Escher,
Variational formulations of steady rotational equatorial waves, Adv. Nonlinear Anal., 10 (2021), 534-547.
doi: 10.1515/anona-2020-0146. |
[9] |
G. M. Coclite and L. d. Ruvo,
Discontinuous solutions for the short-pulse master mode-locking equation, AIMS Math., 4 (2019), 437-462.
doi: 10.3934/math.2019.3.437. |
[10] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[11] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[12] |
A. Constantin and H. P. McKean,
A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982.
doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. |
[13] |
A. Constantin and W. A. Strauss,
Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610.
doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. |
[14] |
J. Escher and Z. Yin,
Well-posedness, blow-up phenomena, and global solutions for the b-equation, J. Reine Angew. Math., 624 (2008), 51-80.
doi: 10.1515/CRELLE.2008.080. |
[15] |
J. Escher, D. Henry, B. Kolev and T. Lyons,
Two-component equations modelling water waves with constant vorticity, Ann. Mat. Pura Appl., 195 (2016), 249-271.
doi: 10.1007/s10231-014-0461-z. |
[16] |
E. D. Farnum and J. N. Kutz,
Dynamics of a low-dimensional model for short pulse mode locking, Photonics, 2 (2015), 865-882.
doi: 10.3390/photonics2030865. |
[17] |
A. S. Fokas and B. Fuchssteiner,
Symplectic structures, their B$\ddot{a}$klund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[18] |
G. L. Gui, Y. Liu and T. X. Tian,
Global existence and blow-up phenomena for the peakon b-family of equations, Indiana Univ. Math. J., 57 (2008), 1209-1234.
doi: 10.1512/iumj.2008.57.3213. |
[19] |
A. Himonas and C. Holliman,
On well-posedness of the Degasperis-Procesi equation, Discrete Contin. Dyn. Syst., 31 (2011), 469-488.
doi: 10.3934/dcds.2011.31.469. |
[20] |
D. D. Holm and M. F. Staley,
Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380.
doi: 10.1137/S1111111102410943. |
[21] |
R. I. Ivanov,
Water waves and integrability, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2267-2280.
doi: 10.1098/rsta.2007.2007. |
[22] |
T. Kato,
Quasi-linear equations of evolution, with applications to partial differential equations in "Spectral theory and differential equations", Lecture Notes in Math., Springer Verlag, Berlin, 448 (1975), 25-70.
|
[23] |
T. Kato,
On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99.
doi: 10.1007/BF01647967. |
[24] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[25] |
M. Li and Z. Yin, Global existence and local well-posedness of the single-cycle pulse equation, J. Math. Phys., 58 (2017), 101515, 16pp.
doi: 10.1063/1.5001381. |
[26] |
W. Luo and Z. Yin,
Gevrey regularity and analyticity for Camassa-Holm type systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1061-1079.
|
[27] |
Y. Liu, D. Pelinovsky and A. Sakovich,
Wave breaking in the short-pulse equation, Dyn. Partial Differ. Equ., 6 (2009), 291-310.
doi: 10.4310/DPDE.2009.v6.n4.a1. |
[28] |
Z. Luo, Z. Qiao and Z. Yin,
On the Cauchy problem for a modified Camassa-Holm equation, Monatsh. Math., 193 (2020), 857-877.
doi: 10.1007/s00605-020-01426-3. |
[29] |
Y. Mi, Y. Liu, B. Guo and T. Luo,
The Cauchy problem for a generalized Camassa-Holm equation, J. Differential Equations, 266 (2019), 6739-6770.
doi: 10.1016/j.jde.2018.11.019. |
[30] |
A. V. Mikhailov and V. S. Novikov,
Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790.
doi: 10.1088/0305-4470/35/22/309. |
[31] |
B. Moon,
Single peaked traveling wave solutions to a generalized $\mu$-Novikov equation, Adv. Nonlinear Anal., 10 (2021), 66-75.
doi: 10.1515/anona-2020-0106. |
[32] |
P. J. Olver and P. Rosenau,
Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E(3), 53 (1996), 1900-1906.
doi: 10.1103/PhysRevE.53.1900. |
[33] |
D. Pelinovsky and A. Sakovich,
Global well-posedness of the short-pulse and sine-Gordon equations in energy space, Comm. Partial Differential Equations, 35 (2010), 613-629.
doi: 10.1080/03605300903509104. |
[34] |
T. Schäfer and C. E. Wayne,
Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D, 196 (2004), 90-105.
doi: 10.1016/j.physd.2004.04.007. |
[35] |
S. Wu, J. Escher and Z. Yin,
Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 633-645.
doi: 10.3934/dcdsb.2009.12.633. |
[36] |
R. Xu and Y. Yang,
Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect, Discrete Contin. Dyn. Syst., 40 (2020), 6507-6527.
doi: 10.3934/dcds.2020288. |
[37] |
K. Yan and Z. Yin,
Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269 (2011), 1113-1127.
doi: 10.1007/s00209-010-0775-5. |
[38] |
M. Yang, Y. Li and Z. Qiao,
Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system, Discrete Contin. Dyn. Syst., 40 (2020), 2475-2493.
doi: 10.3934/dcds.2020122. |
[39] |
Z. Zhaqilao, Q. Hu and Z. Qiao,
Multi-soliton solutions and the Cauchy problem for a two-component short pulse system, Nonlinearity, 30 (2017), 3773-3798.
doi: 10.1088/1361-6544/aa7e9c. |
[40] |
S. Zhou,
Well-posedness and blowup phenomena for a cross-coupled Camassa-Holm equation with waltzing peakons and compacton pairs, J. Evol. Equ., 14 (2014), 727-747.
doi: 10.1007/s00028-014-0236-4. |
[41] |
M. Zhu, Y. Liu and Y. Mi,
Wave-breaking phenomena and persistence properties for the nonlocal rotation-Camassa-Holm equation, Ann. Mat. Pura Appl., 199 (2020), 355-377.
doi: 10.1007/s10231-019-00882-5. |
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