- Previous Article
- DCDS Home
- This Issue
-
Next Article
Global exponential κ-dissipative semigroups and exponential attraction
The Cauchy problem for a generalized Novikov equation
1. | Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China |
2. | Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China, and, Faculty of Information Technology, Macau University of Science and Technology, Macau, China |
We establish the local well-posedness for a generalized Novikov equation in nonhomogeneous Besov spaces. Besides, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.
References:
[1] |
H. Bahouri, J. -Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Springer-Verlag Berlin Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[3] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
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), 1559-1588.
doi: 10.1137/090748500. |
[5] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
G. M. Coclite and K. H. Karlsen,
On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91.
doi: 10.1016/j.jfa.2005.07.008. |
[7] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[8] |
A. Constantin,
Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[9] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[10] |
A. Constantin,
Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.
doi: 10.1093/imamat/hxs033. |
[11] |
A. Constantin and J. Escher,
Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. |
[12] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[13] |
A. Constantin and J. Escher,
Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
doi: 10.1090/S0273-0979-07-01159-7. |
[14] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2001), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[15] |
A. Constantin, V. S. Gerdjikov and R. I. Ivanov,
Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.
doi: 10.1088/0266-5611/22/6/017. |
[16] |
A. Constantin, R. I. Ivanov and J. Lenells,
Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575.
doi: 10.1088/0951-7715/23/10/012. |
[17] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[18] |
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. |
[19] |
A. Constantin and L. Molinet,
Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[20] |
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. |
[21] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
|
[22] |
A. Degasperis, D. D. Holm and A. N. W. Hone,
A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133 (2002), 1463-1474.
doi: 10.1023/A:1021186408422. |
[23] |
A. Degasperis and M. Procesi,
Asymptotic integrability, in Symmetry and perturbation theory (Rome, 1998), World Sci. Publ., River Edge, NJ, 133 (1999), 23-37.
|
[24] |
H. R. Dullin, G. A. Gottwald and D. D. Holm,
On asymptotically equivalent shallow water wave equations, Phys. D, 190 (2004), 1-14.
doi: 10.1016/j.physd.2003.11.004. |
[25] |
J. Escher, Y. Liu and Z. Yin,
Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485.
doi: 10.1016/j.jfa.2006.03.022. |
[26] |
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), 87-117.
doi: 10.1512/iumj.2007.56.3040. |
[27] |
A. Fokas and B. Fuchssteiner,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[28] |
G. Gui and Y. Liu,
On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math., 69 (2011), 445-464.
doi: 10.1090/S0033-569X-2011-01216-5. |
[29] |
G. Gui, Y. Liu, P. J. Olver and C. Qu,
Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759.
doi: 10.1007/s00220-012-1566-0. |
[30] |
A. N. W. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity J. Phys. A 41 (2008), 372002, 10pp.
doi: 10.1088/1751-8113/41/37/372002. |
[31] |
H. He and Z. Yin,
On a generalized Camassa-Holm equation with the flow generated by velocity and its gradient, Appl. Anal., 96 (2017), 679-701.
doi: 10.1080/00036811.2016.1151498. |
[32] |
J. Lenells,
Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82.
doi: 10.1016/j.jmaa.2004.11.038. |
[33] |
J. Li and Z. Yin,
Well-posedness and global existence for a generalized Degasperis-Procesi equation, Nonlinear Anal. Real World Appl., 28 (2016), 72-90.
doi: 10.1016/j.nonrwa.2015.09.003. |
[34] |
J. Li and Z. Yin,
Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143.
doi: 10.1016/j.jde.2016.08.031. |
[35] |
J. Li and Z. Yin,
Well-posedness and analytic solutions of the two-component Euler-Poincaré system, Monatsh Math, (2016), 1-29.
doi: 10.1007/s00605-016-0927-8. |
[36] |
Y. Liu and Z. Yin,
Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
[37] |
Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Procesi equation Int. Math. Res. Not. IMRN (2007), Art. ID rnm117, 22 pp.
doi: 10.1093/imrn/rnm117. |
[38] |
H. Lundmark,
Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198.
doi: 10.1007/s00332-006-0803-3. |
[39] |
W. Luo and Z. Yin,
Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space, Nonlinear Anal., 122 (2015), 1-22.
doi: 10.1016/j.na.2015.03.022. |
[40] |
T. Lyons,
Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107.
doi: 10.3934/dcds.2014.34.3095. |
[41] |
V. Novikov, Generalization of the Camassa-Holm equation J. Phys. A 42 (2009), 342002, 14pp.
doi: 10.1088/1751-8113/42/34/342002. |
[42] |
J. F. Toland,
Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
|
[43] |
X. Tu and Z. Yin,
Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation in the critical Besov space, Nonlinear Anal., 128 (2015), 1-19.
doi: 10.1016/j.na.2015.07.017. |
[44] |
V. O. Vakhnenko and E. J. Parkes,
Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals, 20 (2004), 1059-1073.
doi: 10.1016/j.chaos.2003.09.043. |
[45] |
X. Wu and Z. Yin, Global weak solutions for the Novikov equation J. Phys. A 44 (2011), 055202, 17pp.
doi: 10.1088/1751-8113/44/5/055202. |
[46] |
X. Wu and Z. Yin,
Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707-727.
|
[47] |
X. Wu and Z. Yin,
A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137.
doi: 10.1080/00036811.2011.649735. |
[48] |
Z. Xin and P. Zhang,
On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.
doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. |
[49] |
W. Yan, Y. Li and Y. Zhang,
The Cauchy problem for the integrable Novikov equation, J. Differential Equations Appl., 253 (2012), 298-318.
doi: 10.1016/j.jde.2012.03.015. |
[50] |
W. Yan, Y. Li and Y. Zhang,
The Cauchy problem for the Novikov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1157-1169.
doi: 10.1007/s00030-012-0202-1. |
[51] |
Z. Yin,
On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.
|
[52] |
Z. Yin,
Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139.
doi: 10.1016/S0022-247X(03)00250-6. |
[53] |
Z. Yin,
Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209.
doi: 10.1512/iumj.2004.53.2479. |
[54] |
Z. Yin,
Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194.
doi: 10.1016/j.jfa.2003.07.010. |
show all references
References:
[1] |
H. Bahouri, J. -Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Springer-Verlag Berlin Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
A. Bressan and A. Constantin,
Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239.
doi: 10.1007/s00205-006-0010-z. |
[3] |
A. Bressan and A. Constantin,
Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27.
doi: 10.1142/S0219530507000857. |
[4] |
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), 1559-1588.
doi: 10.1137/090748500. |
[5] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[6] |
G. M. Coclite and K. H. Karlsen,
On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91.
doi: 10.1016/j.jfa.2005.07.008. |
[7] |
A. Constantin,
On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457 (2001), 953-970.
doi: 10.1098/rspa.2000.0701. |
[8] |
A. Constantin,
Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.
doi: 10.5802/aif.1757. |
[9] |
A. Constantin,
The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
doi: 10.1007/s00222-006-0002-5. |
[10] |
A. Constantin,
Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.
doi: 10.1093/imamat/hxs033. |
[11] |
A. Constantin and J. Escher,
Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.
doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. |
[12] |
A. Constantin and J. Escher,
Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.
doi: 10.1007/BF02392586. |
[13] |
A. Constantin and J. Escher,
Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
doi: 10.1090/S0273-0979-07-01159-7. |
[14] |
A. Constantin and J. Escher,
Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2001), 559-568.
doi: 10.4007/annals.2011.173.1.12. |
[15] |
A. Constantin, V. S. Gerdjikov and R. I. Ivanov,
Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207.
doi: 10.1088/0266-5611/22/6/017. |
[16] |
A. Constantin, R. I. Ivanov and J. Lenells,
Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity, 23 (2010), 2559-2575.
doi: 10.1088/0951-7715/23/10/012. |
[17] |
A. Constantin and D. Lannes,
The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.
doi: 10.1007/s00205-008-0128-2. |
[18] |
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. |
[19] |
A. Constantin and L. Molinet,
Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: 10.1007/s002200050801. |
[20] |
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. |
[21] |
R. Danchin,
A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
|
[22] |
A. Degasperis, D. D. Holm and A. N. W. Hone,
A new integrable equation with peakon solutions, Teoret. Mat. Fiz., 133 (2002), 1463-1474.
doi: 10.1023/A:1021186408422. |
[23] |
A. Degasperis and M. Procesi,
Asymptotic integrability, in Symmetry and perturbation theory (Rome, 1998), World Sci. Publ., River Edge, NJ, 133 (1999), 23-37.
|
[24] |
H. R. Dullin, G. A. Gottwald and D. D. Holm,
On asymptotically equivalent shallow water wave equations, Phys. D, 190 (2004), 1-14.
doi: 10.1016/j.physd.2003.11.004. |
[25] |
J. Escher, Y. Liu and Z. Yin,
Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485.
doi: 10.1016/j.jfa.2006.03.022. |
[26] |
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), 87-117.
doi: 10.1512/iumj.2007.56.3040. |
[27] |
A. Fokas and B. Fuchssteiner,
Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66.
doi: 10.1016/0167-2789(81)90004-X. |
[28] |
G. Gui and Y. Liu,
On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math., 69 (2011), 445-464.
doi: 10.1090/S0033-569X-2011-01216-5. |
[29] |
G. Gui, Y. Liu, P. J. Olver and C. Qu,
Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys., 319 (2013), 731-759.
doi: 10.1007/s00220-012-1566-0. |
[30] |
A. N. W. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity J. Phys. A 41 (2008), 372002, 10pp.
doi: 10.1088/1751-8113/41/37/372002. |
[31] |
H. He and Z. Yin,
On a generalized Camassa-Holm equation with the flow generated by velocity and its gradient, Appl. Anal., 96 (2017), 679-701.
doi: 10.1080/00036811.2016.1151498. |
[32] |
J. Lenells,
Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82.
doi: 10.1016/j.jmaa.2004.11.038. |
[33] |
J. Li and Z. Yin,
Well-posedness and global existence for a generalized Degasperis-Procesi equation, Nonlinear Anal. Real World Appl., 28 (2016), 72-90.
doi: 10.1016/j.nonrwa.2015.09.003. |
[34] |
J. Li and Z. Yin,
Remarks on the well-posedness of Camassa-Holm type equations in Besov spaces, J. Differential Equations, 261 (2016), 6125-6143.
doi: 10.1016/j.jde.2016.08.031. |
[35] |
J. Li and Z. Yin,
Well-posedness and analytic solutions of the two-component Euler-Poincaré system, Monatsh Math, (2016), 1-29.
doi: 10.1007/s00605-016-0927-8. |
[36] |
Y. Liu and Z. Yin,
Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.
doi: 10.1007/s00220-006-0082-5. |
[37] |
Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Procesi equation Int. Math. Res. Not. IMRN (2007), Art. ID rnm117, 22 pp.
doi: 10.1093/imrn/rnm117. |
[38] |
H. Lundmark,
Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198.
doi: 10.1007/s00332-006-0803-3. |
[39] |
W. Luo and Z. Yin,
Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space, Nonlinear Anal., 122 (2015), 1-22.
doi: 10.1016/j.na.2015.03.022. |
[40] |
T. Lyons,
Particle trajectories in extreme Stokes waves over infinite depth, Discrete Contin. Dyn. Syst., 34 (2014), 3095-3107.
doi: 10.3934/dcds.2014.34.3095. |
[41] |
V. Novikov, Generalization of the Camassa-Holm equation J. Phys. A 42 (2009), 342002, 14pp.
doi: 10.1088/1751-8113/42/34/342002. |
[42] |
J. F. Toland,
Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.
|
[43] |
X. Tu and Z. Yin,
Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation in the critical Besov space, Nonlinear Anal., 128 (2015), 1-19.
doi: 10.1016/j.na.2015.07.017. |
[44] |
V. O. Vakhnenko and E. J. Parkes,
Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals, 20 (2004), 1059-1073.
doi: 10.1016/j.chaos.2003.09.043. |
[45] |
X. Wu and Z. Yin, Global weak solutions for the Novikov equation J. Phys. A 44 (2011), 055202, 17pp.
doi: 10.1088/1751-8113/44/5/055202. |
[46] |
X. Wu and Z. Yin,
Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707-727.
|
[47] |
X. Wu and Z. Yin,
A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137.
doi: 10.1080/00036811.2011.649735. |
[48] |
Z. Xin and P. Zhang,
On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433.
doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. |
[49] |
W. Yan, Y. Li and Y. Zhang,
The Cauchy problem for the integrable Novikov equation, J. Differential Equations Appl., 253 (2012), 298-318.
doi: 10.1016/j.jde.2012.03.015. |
[50] |
W. Yan, Y. Li and Y. Zhang,
The Cauchy problem for the Novikov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1157-1169.
doi: 10.1007/s00030-012-0202-1. |
[51] |
Z. Yin,
On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.
|
[52] |
Z. Yin,
Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139.
doi: 10.1016/S0022-247X(03)00250-6. |
[53] |
Z. Yin,
Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209.
doi: 10.1512/iumj.2004.53.2479. |
[54] |
Z. Yin,
Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194.
doi: 10.1016/j.jfa.2003.07.010. |
[1] |
Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803 |
[2] |
Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781 |
[3] |
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 and Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111 |
[4] |
Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure and Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501 |
[5] |
Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042 |
[6] |
Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521 |
[7] |
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 and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 |
[8] |
Xinwei Yu, Zhichun Zhai. On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1809-1823. doi: 10.3934/cpaa.2012.11.1809 |
[9] |
Wenjing Zhao. Local well-posedness and blow-up criteria of magneto-viscoelastic flows. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4637-4655. doi: 10.3934/dcds.2018203 |
[10] |
Michael S. Jolly, Anuj Kumar, Vincent R. Martinez. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces. Communications on Pure and Applied Analysis, 2022, 21 (1) : 101-120. doi: 10.3934/cpaa.2021169 |
[11] |
Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493 |
[12] |
Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052 |
[13] |
Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure and Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727 |
[14] |
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 |
[15] |
Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461 |
[16] |
Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605 |
[17] |
Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029 |
[18] |
Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 |
[19] |
Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139 |
[20] |
Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]