January  2017, 37(1): 645-661. doi: 10.3934/dcds.2017027

Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion

1. 

Department of Mathematics, Nanjing Forestry University, Nanjing 210037, China

2. 

Department of Mathematics, University of Electronic Science and Technology of China, Chengdu 611731, China

Received  April 2016 Revised  May 2016 Published  November 2016

Considered herein is the blow-up mechanism to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. The first one is designed for the case when linear dispersion is absent and derive a finite-time blow-up result. The key feature is the ratio between solution and its gradient. The second one handles the general situation when the weak linear dispersion is at present. Fortunately, there exist some conserved quantities that bound the $\|u_x\|_{L^4} $ for the periodic generalized modified Camassa-Holm equation, then the breakdown mechanisms are set up for the general case.

Citation: Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027
References:
[1]

L. Brandolese, Local-in-space criteria for blow-up in shallow water and dispersive rod equations, Comm. Math. Phys., 330 (2014), 401-414. doi: 10.1007/s00220-014-1958-4. Google Scholar

[2]

L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 256 (2014), 3981-3998. doi: 10.1016/j.jde.2014.03.008. Google Scholar

[3]

L. Brandolese and M. F. Cortez, On permanent and breading waves inn hyperelastic rods and rings, J. Funct. Anal., 266 (2014), 6954-6987. doi: 10.1016/j.jfa.2014.02.039. Google Scholar

[4]

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A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[6]

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. Google Scholar

[7]

M. ChenY. LiuC. Qu and S. Zhang, Oscillatio-induced blow-up to the modified Camassa-Holm equation with linear dispersion, Adv. Math., 272 (2015), 225-251. doi: 10.1016/j.aim.2014.12.003. Google Scholar

[8]

K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves Ⅰ, Physica D, 162 (2002), 9-33. doi: 10.1016/S0167-2789(01)00364-5. Google Scholar

[9]

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. Google Scholar

[10]

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. Google Scholar

[11]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar

[12]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[13]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26 (1998), 303-328. Google Scholar

[14]

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. Google Scholar

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2), 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[16]

A. Constantin and H. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. Google Scholar

[20]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3. Google Scholar

[21]

H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. Google Scholar

[22]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (ed. A. Degasperis & G. Gaeta), pp 23-37, World Scientific, Singapore, 1999. Google Scholar

[23]

J. EscherY. 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. Google Scholar

[24]

Y. FuG. L. GuiY. Liu and C. Z. Qu, On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 255 (2013), 1905-1938. doi: 10.1016/j.jde.2013.05.024. Google Scholar

[25]

B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[26]

G. L. GuiY. LiuP. Olver and C. Z. 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. Google Scholar

[27]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), 505-523. Google Scholar

[28]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690. Google Scholar

[29]

J. Lenells, A Variational Approach to the Stability of Periodic Peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2. Google Scholar

[30]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. Google Scholar

[31]

Y. LiuP. OlverC. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Anal. Appl. (Singap.), 12 (2014), 355-368. doi: 10.1142/S0219530514500274. Google Scholar

[32]

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. Google Scholar

[33]

G. Misołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar

[34]

V. Novikov, Generalizations of the Camassa-Holm equation J. Phys. A 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002. Google Scholar

[35]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. Google Scholar

[36]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons J. Math. Phys. , 47 (2006), 112701, 9pp. doi: 10.1063/1.2365758. Google Scholar

[37]

C. Z. QuX. C. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic onlinearity, Comm. Math. Phys., 322 (2013), 967-997. doi: 10.1007/s00220-013-1749-3. Google Scholar

[38]

C. Z. QuY. Fu and Y. Liu, Blow-up solutions and peakons to a generalized μ-Camassa-Holm integrable equation, Comm. Math. Phys., 311 (2014), 375-416. doi: 10.1007/s00220-014-2007-z. Google Scholar

[39]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. Google Scholar

[41]

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. Google Scholar

[42]

Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2000), 1815-1844. doi: 10.1081/PDE-120016129. Google Scholar

[43]

Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Cont. Discrete Impuls. Syst. Ser. A, Math. Anal., 12 (2005), 375-381. Google Scholar

[44]

M. Zhu and S. Zhang, On the blow-up of solutions to the periodic modified integrable Camassa-Holm equation, Discrete Contin. Dyn. Syst., 36 (2016), 2347-2364. doi: 10.3934/dcds.2016.36.2347. Google Scholar

show all references

References:
[1]

L. Brandolese, Local-in-space criteria for blow-up in shallow water and dispersive rod equations, Comm. Math. Phys., 330 (2014), 401-414. doi: 10.1007/s00220-014-1958-4. Google Scholar

[2]

L. Brandolese and M. F. Cortez, Blowup issues for a class of nonlinear dispersive wave equations, J. Differential Equations, 256 (2014), 3981-3998. doi: 10.1016/j.jde.2014.03.008. Google Scholar

[3]

L. Brandolese and M. F. Cortez, On permanent and breading waves inn hyperelastic rods and rings, J. Funct. Anal., 266 (2014), 6954-6987. doi: 10.1016/j.jfa.2014.02.039. Google Scholar

[4]

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. Google Scholar

[5]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[6]

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. Google Scholar

[7]

M. ChenY. LiuC. Qu and S. Zhang, Oscillatio-induced blow-up to the modified Camassa-Holm equation with linear dispersion, Adv. Math., 272 (2015), 225-251. doi: 10.1016/j.aim.2014.12.003. Google Scholar

[8]

K. S. Chou and C. Z. Qu, Integrable equations arising from motions of plane curves Ⅰ, Physica D, 162 (2002), 9-33. doi: 10.1016/S0167-2789(01)00364-5. Google Scholar

[9]

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. Google Scholar

[10]

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. Google Scholar

[11]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. Google Scholar

[12]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[13]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 26 (1998), 303-328. Google Scholar

[14]

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. Google Scholar

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2), 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. Google Scholar

[16]

A. Constantin and H. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. Google Scholar

[20]

A. Constantin and W. A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3. Google Scholar

[21]

H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127 (1998), 193-207. doi: 10.1007/BF01170373. Google Scholar

[22]

A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and perturbation theory (ed. A. Degasperis & G. Gaeta), pp 23-37, World Scientific, Singapore, 1999. Google Scholar

[23]

J. EscherY. 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. Google Scholar

[24]

Y. FuG. L. GuiY. Liu and C. Z. Qu, On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations, 255 (2013), 1905-1938. doi: 10.1016/j.jde.2013.05.024. Google Scholar

[25]

B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[26]

G. L. GuiY. LiuP. Olver and C. Z. 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. Google Scholar

[27]

H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), 505-523. Google Scholar

[28]

S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690. Google Scholar

[29]

J. Lenells, A Variational Approach to the Stability of Periodic Peakons, J. Nonlinear Math. Phys., 11 (2004), 151-163. doi: 10.2991/jnmp.2004.11.2.2. Google Scholar

[30]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. Google Scholar

[31]

Y. LiuP. OlverC. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Anal. Appl. (Singap.), 12 (2014), 355-368. doi: 10.1142/S0219530514500274. Google Scholar

[32]

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. Google Scholar

[33]

G. Misołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7. Google Scholar

[34]

V. Novikov, Generalizations of the Camassa-Holm equation J. Phys. A 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002. Google Scholar

[35]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900. Google Scholar

[36]

Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons J. Math. Phys. , 47 (2006), 112701, 9pp. doi: 10.1063/1.2365758. Google Scholar

[37]

C. Z. QuX. C. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic onlinearity, Comm. Math. Phys., 322 (2013), 967-997. doi: 10.1007/s00220-013-1749-3. Google Scholar

[38]

C. Z. QuY. Fu and Y. Liu, Blow-up solutions and peakons to a generalized μ-Camassa-Holm integrable equation, Comm. Math. Phys., 311 (2014), 375-416. doi: 10.1007/s00220-014-2007-z. Google Scholar

[39]

G. Rodriguez-Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. Google Scholar

[41]

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. Google Scholar

[42]

Z. Xin and P. Zhang, On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 27 (2000), 1815-1844. doi: 10.1081/PDE-120016129. Google Scholar

[43]

Z. Yin, On the blow-up of solutions of the periodic Camassa-Holm equation, Dyn. Cont. Discrete Impuls. Syst. Ser. A, Math. Anal., 12 (2005), 375-381. Google Scholar

[44]

M. Zhu and S. Zhang, On the blow-up of solutions to the periodic modified integrable Camassa-Holm equation, Discrete Contin. Dyn. Syst., 36 (2016), 2347-2364. doi: 10.3934/dcds.2016.36.2347. Google Scholar

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