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, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967
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show all references

References:
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A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., 43 (2001), 585-620. doi: 10.1137/S0036144501386986.  Google Scholar

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R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. 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), 1-33. doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[8]

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[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231.  Google Scholar

[10]

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

[11]

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[12]

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[16]

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[17]

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[18]

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[19]

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

[20]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.  Google Scholar

[21]

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.  Google Scholar

[22]

A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x.  Google Scholar

[23]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.  Google Scholar

[24]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar

[25]

R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November, 2003. Google Scholar

[26]

A. Degasperis and M. Procesi, Asymptotic integrability, in: Symmetry and Perturbation Theory, World Scientific, Singapore, (1999), 23-37.  Google Scholar

[27]

A. Degasperis, D. Holm and A. Hone, A new integral equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474. 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, II (Gallipoli 2002), World Sci. Publ., River Edge, NJ, (2003), 37-43. 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), 73-95. doi: 10.1016/S0169-5983(03)00046-7.  Google Scholar

[30]

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.  Google Scholar

[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), 4501-4504. doi: 10.1103/PhysRevLett.87.194501.  Google Scholar

[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), 457-485. doi: 10.1016/j.jfa.2006.03.022.  Google Scholar

[33]

J. Escher, Y. Liu, 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

[34]

J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the $b$-equation, J. reine Angew. Math., 624 (2008), 51-80. doi: 10.1515/CRELLE.2008.080.  Google Scholar

[35]

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.  Google Scholar

[36]

K. Gröchenig, Weight functions in time-frequency analysis, Pseudo-differential operators: Partial differential equations and time-frequency analysis, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 52 (2007), 343-366.  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), 1209-1234. 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), 1565-1573. 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), 449-479. 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), 511-522. doi: 10.1007/s00220-006-0172-4.  Google Scholar

[41]

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