November  2014, 34(11): 4967-4986. doi: 10.3934/dcds.2014.34.4967

The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces

1. 

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China

Received  August 2013 Revised  March 2014 Published  May 2014

This paper deals with the Cauchy problem for a generalized $b$-equation with higher-order nonlinearities $y_{t}+u^{m+1}y_{x}+bu^{m}u_{x}y=0$, where $b$ is a constant and $m\in\mathbb{N}$, the notation $y:= (1-\partial_x^2) u$, which includes the famous $b$-equation and Novikov equation as special cases. The local well-posedness in critical Besov space $B^{3/2}_{2,1}$ is established. Moreover, a lower bound for the maximal existence time is derived. Finally, the persistence properties in weighted $L^p$ spaces for the solution of this equation are considered, which extend the work of Brandolese [L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not. 22 (2012), 5161-5181] on persistence properties to more general equation with higher-order nonlinearities.
Citation: Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967
References:
[1]

A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces,, SIAM Rev., 43 (2001), 585. doi: 10.1137/S0036144501386986. Google Scholar

[2]

R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy,, Adv. Math., 140 (1998), 190. doi: 10.1006/aima.1998.1768. Google Scholar

[3]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar

[4]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces,, Int. Math. Res. Not., 22 (2012), 5161. Google Scholar

[5]

A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line,, C. R. Math. Acad. Sci. Paris, 343 (2006), 627. doi: 10.1016/j.crma.2006.10.014. Google Scholar

[6]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Letters, 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[7]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0. Google Scholar

[8]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 233 (2006), 60. doi: 10.1016/j.jfa.2005.07.008. Google Scholar

[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352. doi: 10.1006/jfan.1997.3231. Google Scholar

[10]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar

[11]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London, 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar

[12]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 26 (1998), 303. Google Scholar

[13]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar

[14]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana. Univ. Math. J., 47 (1998), 1527. doi: 10.1512/iumj.1998.47.1466. Google Scholar

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[16]

A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197. doi: 10.1088/0266-5611/22/6/017. Google Scholar

[17]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar

[18]

A. Constantin, R. I. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation,, Nonlinearity, 23 (2010), 2559. doi: 10.1088/0951-7715/23/10/012. Google Scholar

[19]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar

[20]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: 10.1007/s002200050801. Google Scholar

[21]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar

[22]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonlinear. Sci., 12 (2002), 415. doi: 10.1007/s00332-002-0517-x. Google Scholar

[23]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953. Google Scholar

[24]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429. doi: 10.1016/S0022-0396(03)00096-2. Google Scholar

[25]

R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003). Google Scholar

[26]

A. Degasperis and M. Procesi, Asymptotic integrability, in: Symmetry and Perturbation Theory,, World Scientific, (1999), 23. Google Scholar

[27]

A. Degasperis, D. Holm and A. Hone, A new integral equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422. Google Scholar

[28]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integral and non-integrable equations with peakons,, Nonlinear physics: Theory and experiment, (2003), 37. doi: 10.1142/9789812704467_0005. Google Scholar

[29]

H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dyn. Res., 33 (2003), 73. doi: 10.1016/S0169-5983(03)00046-7. Google Scholar

[30]

H. R. Dullin, G. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations,, Phys. D., 190 (2004), 1. doi: 10.1016/j.physd.2003.11.004. Google Scholar

[31]

H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Letters, 87 (2001), 4501. doi: 10.1103/PhysRevLett.87.194501. Google Scholar

[32]

J. Escher, Y. Liu and Z. Y. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457. doi: 10.1016/j.jfa.2006.03.022. Google Scholar

[33]

J. Escher, Y. Liu, Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation,, Indiana Univ. Math. J., 56 (2007), 87. doi: 10.1512/iumj.2007.56.3040. Google Scholar

[34]

J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the $b$-equation,, J. reine Angew. Math., 624 (2008), 51. doi: 10.1515/CRELLE.2008.080. Google Scholar

[35]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, Phys. D, 4 (): 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[36]

K. Gröchenig, Weight functions in time-frequency analysis, Pseudo-differential operators: Partial differential equations and time-frequency analysis,, Fields Inst. Commun., 52 (2007), 343. Google Scholar

[37]

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. doi: 10.1512/iumj.2008.57.3213. Google Scholar

[38]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104. Google Scholar

[39]

A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449. doi: 10.1088/0951-7715/25/2/449. Google Scholar

[40]

A. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation,, Comm. Math. Phy., 271 (2007), 511. doi: 10.1007/s00220-006-0172-4. Google Scholar

[41]

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. doi: 10.1137/S1111111102410943. Google Scholar

[42]

D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1-1 nonlinear evolutionary PDE,, Phys. Lett. A, 308 (2003), 437. doi: 10.1016/S0375-9601(03)00114-2. Google Scholar

[43]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. Appl. Math. Theor., 41 (2008). doi: 10.1088/1751-8113/41/37/372002. Google Scholar

[44]

A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation,, Dyn. Partial Differ. Equ., 6 (2009), 253. doi: 10.4310/DPDE.2009.v6.n3.a3. Google Scholar

[45]

R. I. Ivanov, Water waves and integrability,, Philos. Trans. Roy. Soc. London A, 365 (2007), 2267. doi: 10.1098/rsta.2007.2007. Google Scholar

[46]

R. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities,, Z. Naturforsch. A, 61 (2006), 133. Google Scholar

[47]

Z. H. Jiang and L. D. Ni, Blow-up phemomena for the integrable Novikov equation,, J. Math. Appl. Anal., 385 (2012), 551. doi: 10.1016/j.jmaa.2011.06.067. Google Scholar

[48]

K. Grayshan, Peakon solutions of the Novikov equation and properties of the data-to-solution map,, J. Math. Anal. Appl., 397 (2013), 515. doi: 10.1016/j.jmaa.2012.08.006. Google Scholar

[49]

S. Y Lai, N. Li and Y. H. Wu, The existence of global strong and weak solutions for the Novikov equation,, J. Math. Anal. Appl., 399 (2013), 682. doi: 10.1016/j.jmaa.2012.10.048. Google Scholar

[50]

J. Lenells, Conservation laws of the Camassa-Holm equation,, J. Phys. (A), 38 (2005), 869. doi: 10.1088/0305-4470/38/4/007. Google Scholar

[51]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Diff. Equ., 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar

[52]

N. Li, S. Y. Lai, S. Li and M. Wu, The local and global existence of solutions for a generalized Camasa-Holm equation,, Abstr. Appl. Anal., (2012). doi: 10.1155/2012/532369. Google Scholar

[53]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801. doi: 10.1007/s00220-006-0082-5. Google Scholar

[54]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inverse Problems, 19 (2003), 1241. doi: 10.1088/0266-5611/19/6/001. Google Scholar

[55]

A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach,, J. Phys. (A), 35 (2002), 4775. doi: 10.1088/0305-4470/35/22/309. Google Scholar

[56]

L. D. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation,, J. Differential Equations, 250 (2011), 3002. doi: 10.1016/j.jde.2011.01.030. Google Scholar

[57]

L. D. Ni and Y. Zhou, A new asymptotic behavior of solutions to the Camassa-holm equation,, Proc. Amer. Math. Soc., 140 (2012), 607. doi: 10.1090/S0002-9939-2011-10922-5. Google Scholar

[58]

W. Niu and S. Zhang, Blow-up phenomena and global existence for the nouniform weakly dissipative $b$-equation,, J. Math. Anal. Appl., 374 (2011), 166. doi: 10.1016/j.jmaa.2010.08.002. Google Scholar

[59]

V. S. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/34/342002. Google Scholar

[60]

F. Tiglay, The periodic cauchy problem for novikov's equation,, Int. Math. Res. Not., 20 (2011), 4633. doi: 10.1093/imrn/rnq267. Google Scholar

[61]

V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation,, Chaos Solitons Fractals, 20 (2004), 1059. doi: 10.1016/j.chaos.2003.09.043. Google Scholar

[62]

X. L. Wu and Z. Y. Yin, A note on the Cauchy problem of the Novikov equation,, Appl. Anal., 92 (2013), 1116. doi: 10.1080/00036811.2011.649735. Google Scholar

[63]

S. Y. Wu and Z. Y. Yin, Global weak solutions for the Novikov equation,, J. Phys. A: Math. Theor., 44 (2011). doi: 10.1088/1751-8113/44/5/055202. Google Scholar

[64]

Z. P. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar

[65]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation,, J. Differential Equations, 253 (2012), 298. doi: 10.1016/j.jde.2012.03.015. Google Scholar

[66]

Z. Y. Yin, Global solutions to a new integrable equation with peakons,, Indiana. Univ. Math. J., 53 (2004), 1189. doi: 10.1512/iumj.2004.53.2479. Google Scholar

[67]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, IIIinois J. Math., 47 (2003), 649. Google Scholar

[68]

W. Yan, Y. S. Li and Y. M. Zhang, The cauchy problem for the Novikov equation,, Nonlinear Differ. Equ. Appl., 20 (2013), 1157. doi: 10.1007/s00030-012-0202-1. Google Scholar

[69]

S. M. Zhou and C. L. Mu, The properties of solutions for a generalized $b$-family equation with higher-order nonlinearities and peakons,, J. Nonlinear Sci., 23 (2013), 863. doi: 10.1007/s00332-013-9171-8. Google Scholar

[70]

S. M. Zhou, C. L. Mu and L. C. Wang, Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation,, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 843. doi: 10.3934/dcds.2014.34.843. Google Scholar

show all references

References:
[1]

A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces,, SIAM Rev., 43 (2001), 585. doi: 10.1137/S0036144501386986. Google Scholar

[2]

R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy,, Adv. Math., 140 (1998), 190. doi: 10.1006/aima.1998.1768. Google Scholar

[3]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z. Google Scholar

[4]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces,, Int. Math. Res. Not., 22 (2012), 5161. Google Scholar

[5]

A. Boutet de Monvel and D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line,, C. R. Math. Acad. Sci. Paris, 343 (2006), 627. doi: 10.1016/j.crma.2006.10.014. Google Scholar

[6]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Letters, 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[7]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0. Google Scholar

[8]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 233 (2006), 60. doi: 10.1016/j.jfa.2005.07.008. Google Scholar

[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352. doi: 10.1006/jfan.1997.3231. Google Scholar

[10]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757. Google Scholar

[11]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. Roy. Soc. London, 457 (2001), 953. doi: 10.1098/rspa.2000.0701. Google Scholar

[12]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 26 (1998), 303. Google Scholar

[13]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar

[14]

A. Constantin and J. Escher, Global weak solutions for a shallow water equation,, Indiana. Univ. Math. J., 47 (1998), 1527. doi: 10.1512/iumj.1998.47.1466. Google Scholar

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[16]

A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation,, Inverse Problems, 22 (2006), 2197. doi: 10.1088/0266-5611/22/6/017. Google Scholar

[17]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050. Google Scholar

[18]

A. Constantin, R. I. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation,, Nonlinearity, 23 (2010), 2559. doi: 10.1088/0951-7715/23/10/012. Google Scholar

[19]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. Google Scholar

[20]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: 10.1007/s002200050801. Google Scholar

[21]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar

[22]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons,, J. Nonlinear. Sci., 12 (2002), 415. doi: 10.1007/s00332-002-0517-x. Google Scholar

[23]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953. Google Scholar

[24]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429. doi: 10.1016/S0022-0396(03)00096-2. Google Scholar

[25]

R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003). Google Scholar

[26]

A. Degasperis and M. Procesi, Asymptotic integrability, in: Symmetry and Perturbation Theory,, World Scientific, (1999), 23. Google Scholar

[27]

A. Degasperis, D. Holm and A. Hone, A new integral equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422. Google Scholar

[28]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integral and non-integrable equations with peakons,, Nonlinear physics: Theory and experiment, (2003), 37. doi: 10.1142/9789812704467_0005. Google Scholar

[29]

H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dyn. Res., 33 (2003), 73. doi: 10.1016/S0169-5983(03)00046-7. Google Scholar

[30]

H. R. Dullin, G. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations,, Phys. D., 190 (2004), 1. doi: 10.1016/j.physd.2003.11.004. Google Scholar

[31]

H. R. Dullin, G. A. Gottwald and D. D. Holm, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Letters, 87 (2001), 4501. doi: 10.1103/PhysRevLett.87.194501. Google Scholar

[32]

J. Escher, Y. Liu and Z. Y. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457. doi: 10.1016/j.jfa.2006.03.022. Google Scholar

[33]

J. Escher, Y. Liu, Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation,, Indiana Univ. Math. J., 56 (2007), 87. doi: 10.1512/iumj.2007.56.3040. Google Scholar

[34]

J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the $b$-equation,, J. reine Angew. Math., 624 (2008), 51. doi: 10.1515/CRELLE.2008.080. Google Scholar

[35]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries,, Phys. D, 4 (): 47. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[36]

K. Gröchenig, Weight functions in time-frequency analysis, Pseudo-differential operators: Partial differential equations and time-frequency analysis,, Fields Inst. Commun., 52 (2007), 343. Google Scholar

[37]

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. doi: 10.1512/iumj.2008.57.3213. Google Scholar

[38]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104. Google Scholar

[39]

A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449. doi: 10.1088/0951-7715/25/2/449. Google Scholar

[40]

A. A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation,, Comm. Math. Phy., 271 (2007), 511. doi: 10.1007/s00220-006-0172-4. Google Scholar

[41]

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. doi: 10.1137/S1111111102410943. Google Scholar

[42]

D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1-1 nonlinear evolutionary PDE,, Phys. Lett. A, 308 (2003), 437. doi: 10.1016/S0375-9601(03)00114-2. Google Scholar

[43]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. Appl. Math. Theor., 41 (2008). doi: 10.1088/1751-8113/41/37/372002. Google Scholar

[44]

A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation,, Dyn. Partial Differ. Equ., 6 (2009), 253. doi: 10.4310/DPDE.2009.v6.n3.a3. Google Scholar

[45]

R. I. Ivanov, Water waves and integrability,, Philos. Trans. Roy. Soc. London A, 365 (2007), 2267. doi: 10.1098/rsta.2007.2007. Google Scholar

[46]

R. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities,, Z. Naturforsch. A, 61 (2006), 133. Google Scholar

[47]

Z. H. Jiang and L. D. Ni, Blow-up phemomena for the integrable Novikov equation,, J. Math. Appl. Anal., 385 (2012), 551. doi: 10.1016/j.jmaa.2011.06.067. Google Scholar

[48]

K. Grayshan, Peakon solutions of the Novikov equation and properties of the data-to-solution map,, J. Math. Anal. Appl., 397 (2013), 515. doi: 10.1016/j.jmaa.2012.08.006. Google Scholar

[49]

S. Y Lai, N. Li and Y. H. Wu, The existence of global strong and weak solutions for the Novikov equation,, J. Math. Anal. Appl., 399 (2013), 682. doi: 10.1016/j.jmaa.2012.10.048. Google Scholar

[50]

J. Lenells, Conservation laws of the Camassa-Holm equation,, J. Phys. (A), 38 (2005), 869. doi: 10.1088/0305-4470/38/4/007. Google Scholar

[51]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Diff. Equ., 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar

[52]

N. Li, S. Y. Lai, S. Li and M. Wu, The local and global existence of solutions for a generalized Camasa-Holm equation,, Abstr. Appl. Anal., (2012). doi: 10.1155/2012/532369. Google Scholar

[53]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801. doi: 10.1007/s00220-006-0082-5. Google Scholar

[54]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inverse Problems, 19 (2003), 1241. doi: 10.1088/0266-5611/19/6/001. Google Scholar

[55]

A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach,, J. Phys. (A), 35 (2002), 4775. doi: 10.1088/0305-4470/35/22/309. Google Scholar

[56]

L. D. Ni and Y. Zhou, Well-posedness and persistence properties for the Novikov equation,, J. Differential Equations, 250 (2011), 3002. doi: 10.1016/j.jde.2011.01.030. Google Scholar

[57]

L. D. Ni and Y. Zhou, A new asymptotic behavior of solutions to the Camassa-holm equation,, Proc. Amer. Math. Soc., 140 (2012), 607. doi: 10.1090/S0002-9939-2011-10922-5. Google Scholar

[58]

W. Niu and S. Zhang, Blow-up phenomena and global existence for the nouniform weakly dissipative $b$-equation,, J. Math. Anal. Appl., 374 (2011), 166. doi: 10.1016/j.jmaa.2010.08.002. Google Scholar

[59]

V. S. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/34/342002. Google Scholar

[60]

F. Tiglay, The periodic cauchy problem for novikov's equation,, Int. Math. Res. Not., 20 (2011), 4633. doi: 10.1093/imrn/rnq267. Google Scholar

[61]

V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation,, Chaos Solitons Fractals, 20 (2004), 1059. doi: 10.1016/j.chaos.2003.09.043. Google Scholar

[62]

X. L. Wu and Z. Y. Yin, A note on the Cauchy problem of the Novikov equation,, Appl. Anal., 92 (2013), 1116. doi: 10.1080/00036811.2011.649735. Google Scholar

[63]

S. Y. Wu and Z. Y. Yin, Global weak solutions for the Novikov equation,, J. Phys. A: Math. Theor., 44 (2011). doi: 10.1088/1751-8113/44/5/055202. Google Scholar

[64]

Z. P. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar

[65]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation,, J. Differential Equations, 253 (2012), 298. doi: 10.1016/j.jde.2012.03.015. Google Scholar

[66]

Z. Y. Yin, Global solutions to a new integrable equation with peakons,, Indiana. Univ. Math. J., 53 (2004), 1189. doi: 10.1512/iumj.2004.53.2479. Google Scholar

[67]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, IIIinois J. Math., 47 (2003), 649. Google Scholar

[68]

W. Yan, Y. S. Li and Y. M. Zhang, The cauchy problem for the Novikov equation,, Nonlinear Differ. Equ. Appl., 20 (2013), 1157. doi: 10.1007/s00030-012-0202-1. Google Scholar

[69]

S. M. Zhou and C. L. Mu, The properties of solutions for a generalized $b$-family equation with higher-order nonlinearities and peakons,, J. Nonlinear Sci., 23 (2013), 863. doi: 10.1007/s00332-013-9171-8. Google Scholar

[70]

S. M. Zhou, C. L. Mu and L. C. Wang, Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation,, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 843. doi: 10.3934/dcds.2014.34.843. Google Scholar

[1]

Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809

[2]

Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203

[3]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781

[4]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[5]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501

[6]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025

[7]

Tarek Saanouni. A note on global well-posedness and blow-up of some semilinear evolution equations. Evolution Equations & Control Theory, 2015, 4 (3) : 355-372. doi: 10.3934/eect.2015.4.355

[8]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493

[9]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171

[10]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042

[11]

Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365

[12]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053

[13]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803

[14]

Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019

[15]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[16]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112

[17]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327

[18]

P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691

[19]

Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371

[20]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]