February  2014, 34(2): 843-867. doi: 10.3934/dcds.2014.34.843

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331

Received  November 2012 Revised  April 2013 Published  August 2013

This paper deals with the Cauchy problem for a weakly dissipative shallow water equation with high-order nonlinearities $y_{t}+u^{m+1}y_{x}+bu^{m}u_{x}y+\lambda y=0$, where $\lambda,b$ are constants and $m\in\mathbb{N}$, the notation $y:= (1-\partial_x^2) u$, which includes the famous $b$-equation and Novikov equations as special cases. The local well-posedness of solutions for the Cauchy problem in Besov space $B^s_{p,r} $ with $1\leq p,r \leq +\infty$ and $s>\max\{1+\frac{1}{p},\frac{3}{2}\}$ is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are acquired, moreover, the propagation behaviors of compactly supported solutions are also established. Finally, the weak solution and analytic solution for the equation are considered.
Citation: 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
References:
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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]

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.

[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.

[4]

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.

[5]

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.

[6]

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.

[7]

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.

[8]

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

[9]

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.

[10]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701.

[11]

A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[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.

[23]

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

[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.

[25]

R. Danchin, "Fourier Analysis Methods for PDEs,", Lecture Notes, 14 (2003).

[26]

A. Degasperis and M. Procesi, Asymptotic integrability,, in, (1999), 23.

[27]

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

[28]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integral and non-integrable equations with peakons,, in, (2003), 37. doi: 10.1142/9789812704467_0005.

[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.

[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.

[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). doi: 10.1103/PhysRevLett.87.194501.

[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.

[33]

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.

[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.

[35]

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,, preprint, ().

[36]

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.

[37]

J.-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time,, J. Differential Equations, 74 (1988), 369. doi: 10.1016/0022-0396(88)90010-1.

[38]

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.

[39]

D. Henry, Compactly supported solutions of the Camassa-Holm equation,, J. Nonlinear Math. Phys., 12 (2005), 342. doi: 10.2991/jnmp.2005.12.3.3.

[40]

D. Henry, Infinite propagation speed for the Degasperis-Procesi equation,, J. Math. Anal. Appl., 311 (2005), 755. doi: 10.1016/j.jmaa.2005.03.001.

[41]

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