The Cauchy problem for a generalized b-equation with higher-order nonlinearities in critical Besov spaces and weighted L^p spaces

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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

Abstract
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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 and Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967

References:

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show all references

References:

[1]	A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., 43 (2001), 585-620. doi: 10.1137/S0036144501386986.
[2]	R. Beals, D. Sattinger and J. Szmigielski, Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math., 140 (1998), 190-206. 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), 215-239. doi: 10.1007/s00205-006-0010-z.
[4]	L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Not., 22 (2012), 5161-5181.
[5]	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), 627-632. doi: 10.1016/j.crma.2006.10.014.
[6]	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.
[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.
[8]	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.
[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.
[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.
[11]	A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.
[12]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 26 (1998), 303-328.
[13]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243. doi: 10.1007/BF02392586.
[14]	A. Constantin and J. Escher, Global weak solutions for a shallow water equation, Indiana. Univ. Math. J., 47 (1998), 1527-1545. 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), 559-568. 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), 2197-2207. 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), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.
[18]	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.
[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.
[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.
[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.
[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.
[23]	R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
[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.
[25]	R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November, 2003.
[26]	A. Degasperis and M. Procesi, Asymptotic integrability, in: Symmetry and Perturbation Theory, World Scientific, Singapore, (1999), 23-37.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X.
[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.
[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.
[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.
[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.
[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.
[41]	D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380. doi: 10.1137/S1111111102410943.
[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), 437-444. doi: 10.1016/S0375-9601(03)00114-2.
[43]	A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. Appl. Math. Theor., 41 (2008), 372002, 10pp. doi: 10.1088/1751-8113/41/37/372002.
[44]	A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289. doi: 10.4310/DPDE.2009.v6.n3.a3.
[45]	R. I. Ivanov, Water waves and integrability, Philos. Trans. Roy. Soc. London A, 365 (2007), 2267-2280. doi: 10.1098/rsta.2007.2007.
[46]	R. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities, Z. Naturforsch. A, 61 (2006), 133-138.
[47]	Z. H. Jiang and L. D. Ni, Blow-up phemomena for the integrable Novikov equation, J. Math. Appl. Anal., 385 (2012), 551-558. doi: 10.1016/j.jmaa.2011.06.067.
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