DCDS
The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces
Shouming Zhou
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.
DCDS
Global conservative and dissipative solutions of the generalized Camassa-Holm equation
Shouming Zhou Chunlai Mu
This paper is devoted to the continuation of solutions to the generalized Camassa-Holm equation beyond wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear system. This formulation allows one to continue the solution after collision time, giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global conservative or dissipative solutions, which depend continuously on the initial data.
keywords: dissipative solutions. conservative solutions Generalized Camassa-Holm equation
DCDS-B
New results of the ultimate bound on the trajectories of the family of the Lorenz systems
Fuchen Zhang Chunlai Mu Shouming Zhou Pan Zheng
In this paper, the global exponential attractive sets of a class of continuous-time dynamical systems defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^3},$ are studied. The elements of main diagonal of matrix $A$ are both negative numbers and zero, where matrix $A$ is the Jacobian matrix $\frac{{df}}{{dx}}$ of a continuous-time dynamical system defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^3},$ evaluated at the origin ${x_0} = \left( {0,0,0} \right).$ The former equations [1-6] that we are searching for a global bounded region have a common characteristic: The elements of main diagonal of matrix $A$ are all negative, where matrix $A$ is the Jacobian matrix $\frac{{df}}{{dx}}$ of a continuous-time dynamical system defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^n},$ evaluated at the origin ${x_0} = {\left( {0,0, \cdots ,0} \right)_{1 \times n}}.$ For the reason that the elements of main diagonal of matrix $A$ are both negative numbers and zero for this class of dynamical systems , the method for constructing the Lyapunov functions that applied to the former dynamical systems does not work for this class of dynamical systems. We overcome this difficulty by adding a cross term $xy$ to the Lyapunov functions of this class of dynamical systems and get a perfect result through many integral inequalities and the generalized Lyapunov functions.
keywords: global attractive set Lyapunov stability generalized Lyapunov functions. Dynamical systems
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
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
Conserved quantities, global existence and blow-up for a generalized CH equation
Long Wei Zhijun Qiao Yang Wang Shouming Zhou

In this paper, we study conserved quantities, blow-up criterions, and global existence of solutions for a generalized CH equation. We investigate the classification of self-adjointness, conserved quantities for this equation from the viewpoint of Lie symmetry analysis. Then, based on these conserved quantities, blow-up criterions and global existence of solutions are presented.

keywords: Nonlinear self-adjointness conservation law global existence blow-up
DCDS
Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation
Shouming Zhou Chunlai Mu Liangchen Wang
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.
keywords: Novikov equation global existence blowup. b-equation well-posedness

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