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

Shouming Zhou; Chunlai Mu; Liangchen Wang

doi:10.3934/dcds.2014.34.843

Article Contents

2014, Volume 34, Issue 2: 843-867. Doi: 10.3934/dcds.2014.34.843

This issue Previous Article A nonlinear diffusion problem arising in population genetics Next Article Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations

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: October 31, 2012

Revised: March 31, 2013

Published: January 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 35G25, 35L05; Secondary: 35Q50.

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