May  2015, 35(5): 2011-2039. doi: 10.3934/dcds.2015.35.2011

A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity

1. 

School of Mathematics, Northwest University, Shaanxi 710127, China

Received  June 2014 Revised  September 2014 Published  December 2014

Considered herein is the Cauchy problem for a modified Camassa-Holm equation with cubic nonlinearity. The local well-posedness in Besov space $B^s_{2,1}$ with the critical index $s=5/2$ is established. Then a lower bound for the maximal time of existence of its solutions is found. With analytic initial data, the solutions to this Cauchy problem are analytic in both variables, globally in space and locally in time, which extends the result of Himonas and Misiołek [A. Himonas, G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann. 327 (2003) 575---584] to more general $\mu$-version equations and systems.
Citation: Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011
References:
[1]

M. S. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems,, J. Differential Equations, 48 (1983), 241. doi: 10.1016/0022-0396(83)90051-7.

[2]

A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation,, SIAM J. Math. Anal., 41 (2009), 1559. doi: 10.1137/090748500.

[3]

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

[4]

J. Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase space analysis of partial differential equations,, Pubbl. Cent. Ric. Mat. Ennio Giorgi, I (2004), 53.

[5]

K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves,, Physica D, 162 (2002), 9. doi: 10.1016/S0167-2789(01)00364-5.

[6]

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.

[7]

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

[8]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.

[9]

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

[10]

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

[11]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75. doi: 10.1007/PL00004793.

[12]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[13]

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

[14]

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

[15]

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.

[16]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[17]

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

[18]

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.

[19]

A. Degasperis and M. Procesi, Asymptotic integrability,, Symmetry and perturbation theory (Rome 1998), (1999), 23.

[20]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons,, Nonlinear physics: Theory and experiment (Gallipoli 2002), II (2003), 37. doi: 10.1142/9789812704467_0005.

[21]

J. Escher, Y. Liu and Z. 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.

[22]

J. Escher, Y. Liu and 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.

[23]

Y. Fu, G. L. Gui, Y. Liu and C. Z. Qu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity,, J. Differential Equations, 255 (2013), 1905. doi: 10.1016/j.jde.2013.05.024.

[24]

Y. Fu, Y. Liu and C. Z. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, J. Funct. Anal., 262 (2012), 3125. doi: 10.1016/j.jfa.2012.01.009.

[25]

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

[26]

G. L. Gui, Y. Liu, P. J. Olver and C. Z. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation,, Comm. Math. Phys., 319 (2013), 731. doi: 10.1007/s00220-012-1566-0.

[27]

A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575. doi: 10.1007/s00208-003-0466-1.

[28]

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

[29]

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

[30]

J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075.

[31]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224.

[32]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Spectral Theory and Differential Equations, 448 (1975), 25.

[33]

B. Khesin, J. Lenells and G. Misiołek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617. doi: 10.1007/s00208-008-0250-3.

[34]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690.

[35]

J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129. doi: 10.1007/s00220-010-1069-9.

[36]

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

[37]

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.

[38]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169. doi: 10.1007/s00332-006-0803-3.

[39]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7.

[40]

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

[41]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900.

[42]

C. Z. Qu, Y. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation,, J. Funct. Anal., 266 (2014), 433. doi: 10.1016/j.jfa.2013.09.021.

[43]

T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media,, Physica D, 196 (2004), 90. doi: 10.1016/j.physd.2004.04.007.

[44]

J. Schiff, The Camassa-Holm equation: A loop group approach,, Physica D, 121 (1998), 24. doi: 10.1016/S0167-2789(98)00099-2.

[45]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.

[46]

F. Tiǧlay, The periodic Cauchy problem of the modified Hunter-Saxton equation,, J. Evol. Equ., 5 (2005), 509. doi: 10.1007/s00028-005-0215-x.

[47]

F. Tiǧlay, The periodic Cauchy problem for Novikov's equation,, Int. Math. Res. Not., 2011 (2011), 4633. doi: 10.1093/imrn/rnq267.

show all references

References:
[1]

M. S. Baouendi and C. Goulaouic, Sharp estimates for analytic pseudodifferential operators and application to the Cauchy problems,, J. Differential Equations, 48 (1983), 241. doi: 10.1016/0022-0396(83)90051-7.

[2]

A. Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschl, Long-time asymptotics for the Camassa-Holm equation,, SIAM J. Math. Anal., 41 (2009), 1559. doi: 10.1137/090748500.

[3]

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

[4]

J. Y. Chemin, Localization in Fourier space and Navier-Stokes system, Phase space analysis of partial differential equations,, Pubbl. Cent. Ric. Mat. Ennio Giorgi, I (2004), 53.

[5]

K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves,, Physica D, 162 (2002), 9. doi: 10.1016/S0167-2789(01)00364-5.

[6]

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.

[7]

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

[8]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.

[9]

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

[10]

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

[11]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation,, Math. Z., 233 (2000), 75. doi: 10.1007/PL00004793.

[12]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[13]

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

[14]

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

[15]

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.

[16]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52 (1999), 949. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D.

[17]

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

[18]

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.

[19]

A. Degasperis and M. Procesi, Asymptotic integrability,, Symmetry and perturbation theory (Rome 1998), (1999), 23.

[20]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons,, Nonlinear physics: Theory and experiment (Gallipoli 2002), II (2003), 37. doi: 10.1142/9789812704467_0005.

[21]

J. Escher, Y. Liu and Z. 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.

[22]

J. Escher, Y. Liu and 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.

[23]

Y. Fu, G. L. Gui, Y. Liu and C. Z. Qu, On the Cauchy problem for the integrable Camassa-Holm type equation with cubic nonlinearity,, J. Differential Equations, 255 (2013), 1905. doi: 10.1016/j.jde.2013.05.024.

[24]

Y. Fu, Y. Liu and C. Z. Qu, On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, J. Funct. Anal., 262 (2012), 3125. doi: 10.1016/j.jfa.2012.01.009.

[25]

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

[26]

G. L. Gui, Y. Liu, P. J. Olver and C. Z. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation,, Comm. Math. Phys., 319 (2013), 731. doi: 10.1007/s00220-012-1566-0.

[27]

A. A. Himonas and G. Misiołek, Analyticity of the Cauchy problem for an integrable evolution equation,, Math. Ann., 327 (2003), 575. doi: 10.1007/s00208-003-0466-1.

[28]

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

[29]

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

[30]

J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075.

[31]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224.

[32]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in Spectral Theory and Differential Equations, 448 (1975), 25.

[33]

B. Khesin, J. Lenells and G. Misiołek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617. doi: 10.1007/s00208-008-0250-3.

[34]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group,, J. Math. Phys., 40 (1999), 857. doi: 10.1063/1.532690.

[35]

J. Lenells, G. Misiołek and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions,, Comm. Math. Phys., 299 (2010), 129. doi: 10.1007/s00220-010-1069-9.

[36]

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

[37]

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.

[38]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169. doi: 10.1007/s00332-006-0803-3.

[39]

G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group,, J. Geom. Phys., 24 (1998), 203. doi: 10.1016/S0393-0440(97)00010-7.

[40]

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

[41]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,, Phys. Rev. E, 53 (1996), 1900. doi: 10.1103/PhysRevE.53.1900.

[42]

C. Z. Qu, Y. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation,, J. Funct. Anal., 266 (2014), 433. doi: 10.1016/j.jfa.2013.09.021.

[43]

T. Schäfer and C. E. Wayne, Propagation of ultra-short optical pulses in cubic nonlinear media,, Physica D, 196 (2004), 90. doi: 10.1016/j.physd.2004.04.007.

[44]

J. Schiff, The Camassa-Holm equation: A loop group approach,, Physica D, 121 (1998), 24. doi: 10.1016/S0167-2789(98)00099-2.

[45]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.

[46]

F. Tiǧlay, The periodic Cauchy problem of the modified Hunter-Saxton equation,, J. Evol. Equ., 5 (2005), 509. doi: 10.1007/s00028-005-0215-x.

[47]

F. Tiǧlay, The periodic Cauchy problem for Novikov's equation,, Int. Math. Res. Not., 2011 (2011), 4633. doi: 10.1093/imrn/rnq267.

[1]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555

[15]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic & Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395

[16]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819

[17]

Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737

[18]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[19]

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

[20]

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

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]