American Institute of Mathematical Sciences

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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

Shouming Zhou 1,

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.
Keywords: local well-posedness, blow-up, Persistence properties, weighted $L^p$ spaces..
Mathematics Subject Classification: Primary: 35G25, 35L05; Secondary: 35Q5.
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:
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show all references

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[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]

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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]

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[11]

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[12]

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[18]

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[21]

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[22]

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[23]

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[24]

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[25]

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[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

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[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]

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[33]

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[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]

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[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]

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