DCDS-B
Dynamical behaviors of a generalized Lorenz family
Fuchen Zhang Xiaofeng Liao Guangyun Zhang Chunlai Mu Min Xiao Ping Zhou
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3707-3720 doi: 10.3934/dcdsb.2017184

In this paper, the ultimate bound set and globally exponentially attractive set of a generalized Lorenz system are studied according to Lyapunov stability theory and optimization theory. The method of constructing Lyapunov-like functions applied to the former Lorenz-type systems (see, e.g. Lorenz system, Rossler system, Chua system) isn't applicable to this generalized Lorenz system. We overcome this difficulty by adding a cross term to the Lyapunov-like functions that used for the Lorenz system to study this generalized Lorenz system. The authors in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications 323 (2006) 844-853] obtained the ultimate bound set of this generalized Lorenz system but only for some cases with $0 ≤ α < \frac{1}{{29}}.$ The ultimate bound set and globally exponential attractive set of this generalized Lorenz system are still unknown for $\alpha \notin \left[ {0, \frac{1}{{29}}} \right).$ Comparing with the best results in the current literature [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, Journal of Mathematical Analysis and Applications 323 (2006) 844-853], our new results fill up the gap of the estimate for the case of $\frac{1}{{29}} ≤ α < \frac{{14}}{{173}}.$ Furthermore, the estimation derived here contains the results given in [D. Li, J. Lu, X. Wu, G. Chen, Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system, J. Math. Anal. Appl. 323 (2006) 844-853] as special case for the case of $0 ≤ α < \frac{1}{{29}}.$

keywords: Generalized Lorenz system Lyapunov-like functions ultimate boundedness global attractive sets
DCDS
Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics
Li Yang Zeng Rong Shouming Zhou Chunlai Mu
Discrete & Continuous Dynamical Systems - A 2018, 38(10): 5205-5220 doi: 10.3934/dcds.2018230

It was showed that the generalized Camassa-Holm equation possible development of singularities in finite time, and beyond the occurrence of wave breaking which exists either global conservative or dissipative solutions. In present paper, we will further investigate the uniqueness of global conservative solutions to it based on the characteristics. From a given conservative solution $u = u(t,x)$, an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities $u$ and $v = 2 \arctan u_x$ along each characteristic, it is obtained that the Cauchy problem with general initial data $u_0∈ H^1(\mathbb{R})$ has a unique global conservative solution.

keywords: The generalized Camassa-Holm equation global conservative solutions uniqueness
DCDS
Global dynamics in a two-species chemotaxis-competition system with two signals
Xinyu Tu Chunlai Mu Pan Zheng Ke Lin
Discrete & Continuous Dynamical Systems - A 2018, 38(7): 3617-3636 doi: 10.3934/dcds.2018156
In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type
$\begin{eqnarray*}\label{1}\left\{\begin{array}{llll}u_t = Δ u-χ_{1}\nabla·(u\nabla v)+μ_{1}u(1-u-a_{1}w), &x∈ Ω, ~~~t>0, \\0 = Δ v-v+w, &x∈Ω, ~~~t>0, \\w_t = Δ w-χ_{2}\nabla·(w\nabla z)+μ_{2}w(1-a_{2}u-w), &x∈ Ω, ~~~ t>0, \\0 = Δ z-z+u, &x∈Ω, ~~~t>0, \\\end{array}\right.\end{eqnarray*}$
with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain
$Ω\subset R^n$
,
$n≥2$
, where
$χ_{i}$
,
$μ_{i}$
and
$a_{i}$
$(i = 1, 2)$
are positive constants. It is shown that for any positive parameters
$χ_{i}$
,
$μ_{i}$
,
$a_{i}$
$(i = 1, 2)$
and any suitably regular initial data
$(u_{0}, w_{0})$
, this system possesses a global bounded classical solution provided that
$\frac{χ_{i}}{μ_{i}}$
are small. Moreover, when
$a_{1}, a_{2}∈ (0, 1)$
and the parameters
$μ_{1}$
and
$μ_{2}$
are sufficiently large, it is proved that the global solution
$(u, v, w, z)$
of this system exponentially approaches to the steady state
$\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$
in the norm of
$L^{∞}(Ω)$
as
$t\to ∞$
; If
$a_{1}≥1>a_{2}>0$
and
$μ_{2}$
is sufficiently large, the solution of the system converges to the constant stationary solution
$\left(0, 1, 1, 0\right)$
as time tends to infinity, and the convergence rates can be calculated accurately.
keywords: Chemotaxis Lotka-Volterra-type competition boundedness Logistic source asymptotic stability
DCDS-B
Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source
Ke Lin Chunlai Mu
Discrete & Continuous Dynamical Systems - B 2017, 22(6): 2233-2260 doi: 10.3934/dcdsb.2017094
In this paper, we consider a system of three parabolic equations in high-dimensional smoothly bounded domain
$\left\{\begin{array}{llll}u_t=\Delta u-\chi_1\nabla\cdot( u\nabla w)+\mu_1u(1-u-a_1v),\quad &x\in \Omega,\quad t>0,\\v_t=\Delta v-\chi_2\nabla\cdot( v\nabla w)+\mu_2v(1-a_2u-v),\quad &x\in\Omega,\quad t>0,\\w_t=\Delta w- w+u+v,\quad &x\in\Omega,\quad t>0,\\\end{array}\right.$
which describes the mutual competition between two populations on account of the Lotka-Volterra dynamics.
For any cross-diffusivities $\chi_1>0$ and $\chi_2>0$ and the rates $a_1>0$ and $a_2>0$, it is proved that the global classical bounded solutions exist for sufficiently regular initial data when the parameters $\mu_1$ and $\mu_2$ are sufficiently large. In deriving the convergence of solutions to this system, we need to distinguish two cases $a_1, a_2\in[0, 1)$ and $a_1>1$ and $0\leq a_2 < 1$ to prove globally asymptotic stability.
keywords: Chemotaxis logistic source boundedness asymptotic stability
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
DCDS
Global conservative and dissipative solutions of the generalized Camassa-Holm equation
Shouming Zhou Chunlai Mu
Discrete & Continuous Dynamical Systems - A 2013, 33(4): 1713-1739 doi: 10.3934/dcds.2013.33.1713
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
Boundedness in a three-dimensional Keller-Segel- Stokes system involving tensor-valued sensitivity with saturation
Dan Li Chunlai Mu Pan Zheng Ke Lin
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-19 doi: 10.3934/dcdsb.2018209
This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with logistic source
$\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, &x∈ Ω, &t>0, \\c_{t}+u·\nabla c = Δ c-c+n, &x∈Ω, &t>0, \\u_{t}+\nabla P = Δ u+n\nablaφ, &x∈Ω, &t>0, \\\nabla· u = 0, &x∈Ω, &t>0\end{array}\right.\end{eqnarray*}$
in three-dimensional smoothly bounded domains, where the parameters $ξ\ge0$, $μ>0$ and $φ∈ W^{1, ∞}(Ω)$, $D$ is a given function satisfying $D(n)\ge C_{D}n^{m-1}$ for all $n>0$ with $m>0$ and $C_{D}>0$. $\mathcal{S}$ is a given function with values in $\mathbb{R}^{3×3}$ which fulfills
$ \begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$
with some $C_{\mathcal{S}}>0$ and $α>0$. It is proved that for all reasonably regular initial data, global weak solutions exist whenever $m+2α>\frac{6}{5}$. This extends a recent result by Liu el at. (J. Diff. Eqns, 261 (2016) 967-999) which asserts global existence of weak solutions under the constraints $m+α>\frac{6}{5}$ and $m\ge\frac{1}{3}$.
keywords: Chemotaxis- Stokes nonlinear diffusion tensor-valued sensitivity boundedness logistic source
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
Discrete & Continuous Dynamical Systems - B 2015, 20(4): 1261-1276 doi: 10.3934/dcdsb.2015.20.1261
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
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
CPAA
Blowup behaviors for degenerate parabolic equations coupled via nonlinear boundary flux
Chunlai Mu Zhaoyin Xiang
Communications on Pure & Applied Analysis 2007, 6(2): 487-503 doi: 10.3934/cpaa.2007.6.487
This paper deals with the blow-up properties of solutions to a degenerate parabolic system coupled via nonlinear boundary flux. Firstly, we construct the self-similar supersolution and subsolution to obtain the critical global existence curve. Secondly, we establish the precise blow-up rate estimates for solutions which blow up in a finite time. Finally, we investigate the localization of blow-up points. The critical curve of Fujita type is conjectured with the aid of some new results.
keywords: Degenerate parabolic system critical global existence curve critical Fujita curve blow-up rate estimates blow-up sets.

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