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### Open Access Journals

DCDS

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.

DCDS

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.

DCDS-B

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.

CPAA

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.

DCDS

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.

DCDS

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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