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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References:
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[8]

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[10]

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[11]

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[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.   Google Scholar

[13]

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[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.  Google Scholar

[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.  Google Scholar

[16]

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[17]

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[18]

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[19]

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[20]

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[21]

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[22]

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[27]

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[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.  Google Scholar

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