The bifurcations of solitary and kink waves described by the Gardner equation
Yiren Chen Zhengrong Liu
Discrete & Continuous Dynamical Systems - S 2016, 9(6): 1629-1645 doi: 10.3934/dcdss.2016067
In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation $u_{t}+a u u_{x}+b u^{2} u_{x}+\gamma u_{xxx}=0$. We obtain some new results as follows: For arbitrary given parameters $b$ and $\gamma$, we choose the parameter $a$ as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.
keywords: kink waves. solitary waves The Gardner equation bifurcations phase analysis
On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces
Hao Tang Zhengrong Liu
Discrete & Continuous Dynamical Systems - A 2016, 36(4): 2229-2256 doi: 10.3934/dcds.2016.36.2229
In this paper, motivated by [13], we use the Littlewood-Paley theory to investigate the Cauchy problem of the Boltzmann equation. When the initial data is a small perturbation of an equilibrium state, under the Grad's angular cutoff assumption, we obtain the unique global strong solution to the Boltzmann equation for the hard potential case in the Chemin-Lerner type spaces $C([0,\infty);\widetilde{L}^{2}_{\xi}(B_{2,r}^{s}))$ with $1\leq r\leq2$ and $s>3/2$ or $s=3/2$ and $r=1$. Besides, we also prove the Lipschitz continuity of the solution map. Our results extend some previous works on the Boltzmann equation in Sobolev spaces.
keywords: Boltzmann equation Littlewood-Paley theory. angular cutoff assumption Cauchy problem hard potential
Periodic solutions of a class of Newtonian equations
Dixiang Cheng Zhengrong Liu Xin Huang
Communications on Pure & Applied Analysis 2009, 8(6): 1795-1801 doi: 10.3934/cpaa.2009.8.1795
In this paper, we apply the Mountain Pass Lemma of Ambrosetti-Rabinowitz [2] to study the existence of new periodic solutions with a prescribed energy for a class of second order Hamiltonian conservative systems.
keywords: periodic solutions Mountain-Pass Lemma. Newtonian equations
Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source
Shijie Shi Zhengrong Liu Hai-Yang Jin
Kinetic & Related Models 2017, 10(3): 855-878 doi: 10.3934/krm.2017034
In this paper, we study an attraction-repulsion Keller-Segel chemotaxis model with logistic source
$\begin{cases} u_{t}=Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w)+f(u), &x∈Ω,\ t>0,\\ v_{t}=Δ v+α u-β v, &x∈Ω,\ t>0,\\ w_{t}=Δ w+γ u-δ w, &x∈Ω,\ t>0,\\\end{cases}\;\;\;\;(*)$
in a smooth bounded domain
$Ω \subset \mathbb{R}^n(n≥ 1)$
, with homogeneous Neumann boundary conditions and nonnegative initial data
satisfying suitable regularity, where
$χ≥ 0,ξ≥ 0,α, β, γ, δ>0$
is a smooth growth source satisfying
$f(0)≥ 0$
$f(u)≤ a-bu^θ, \ \ u≥ 0,\ \ \mathrm{with~some} \ \ a≥ 0,b>0,θ≥1.$
(i.e. repulsion cancels attraction), the boundedness of classical solution of system (*) is established if the dampening parameter
and the space dimension
$\begin{cases} θ > \max\{1,3-\frac6n\}, &\text{when }\ \ 1≤ n≤ 5,\\ θ≥ 2, &\text{when }\ \ 6≤ n≤ 9,\\ θ>1+\frac{2(n-4)}{n+2}, &\text{when} \ \ \ n≥10.\\\end{cases}$
Furthermore, when
$f(u)=μ u(1-u)$
and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if
, the solution
exponentially stabilizes to the constant stationary solution
in the case of
$1≤ n≤ 9$
. Our results implies that when repulsion cancels attraction the logistic source play an important role on the solution behavior of the attraction-repulsion chemotaxis system.
keywords: Chemotaxis boundedness large time behavior attraction-repulsion logistic source

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