KRM
On a three-Component Camassa-Holm equation with peakons
Yongsheng Mi Chunlai Mu
Kinetic & Related Models 2014, 7(2): 305-339 doi: 10.3934/krm.2014.7.305
In this paper, we are concerned with three-Component Camassa-Holm equation with peakons. First, We establish the local well-posedness in a range of the Besov spaces $B^{s}_{p,r},p,r\in [1,\infty],s>\mathrm{ max}\{\frac{3}{2},1+\frac{1}{p}\}$ (which generalize the Sobolev spaces $H^{s}$) by using Littlewood-Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with $s=\frac{3}{2}, p=2,r=1$) is considered. Then, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.
keywords: Besov spaces local well-posedness. Camassa-Holm type equation
CPAA
Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition
Shouming Zhou Chunlai Mu Yongsheng Mi Fuchen Zhang
Communications on Pure & Applied Analysis 2013, 12(6): 2935-2946 doi: 10.3934/cpaa.2013.12.2935
This paper deals with the blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. The local existence and uniqueness of the solution are obtained. Furthermore, we prove that the solution of the equation blows up in finite time. Under appropriate hypotheses, we give the estimates of the blow-up rate, and obtain that the blow-up set is a single point $x=0$ for radially symmetric solution with a single maximum at the origin. Finally, some numerical experiments are performed, which illustrate our results.
keywords: Nonlocal diffusion equation blow-up single point blow-up. blow-up rates
DCDS
On an $N$-Component Camassa-Holm equation with peakons
Yongsheng Mi Boling Guo Chunlai Mu
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1575-1601 doi: 10.3934/dcds.2017065

In this paper, we are concerned with $N$-Component Camassa-Holm equation with peakons. Firstly, we establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory. Secondly, we present a precise blowup scenario and several blowup results for strong solutions to that system, we then obtain the blowup rate of strong solutions when a blowup occurs. Next, we investigate the persistence property for the strong solutions. Finally, we consider the initial boundary value problem, our approach is based on sharp extension results for functions on the half-line and several symmetry preserving properties of the equations under discussion.

keywords: Besov spaces peakon solutions local well-posedness

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