DCDS
Existence, uniqueness, and stability of periodic solutions of an equation of duffing type
Hongbin Chen Yi Li
Discrete & Continuous Dynamical Systems - A 2007, 18(4): 793-807 doi: 10.3934/dcds.2007.18.793
We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically stable with sharp rate of exponential decay. In particular, for a restoring term independent of the variable $t$, a necessary and sufficient condition is obtained which guarantees the existence and uniqueness of a periodic solution that is stable.
keywords: stability. topological degree Periodic solution
DCDS-B
Asymptotic behavior of linearized viscoelastic flow problem
Yinnian He Yi Li
Discrete & Continuous Dynamical Systems - B 2008, 10(4): 843-856 doi: 10.3934/dcdsb.2008.10.843
In this article, we provide some asymptotic behaviors of linearized viscoelastic flows in a general two-dimensional domain with certain parameters small and the time variable large.
keywords: Viscoelastic flows; Navier-Stokes flows; Euler flows; Asymptotic behavior.
DCDS
Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations
Xiaojie Hou Yi Li
Discrete & Continuous Dynamical Systems - A 2006, 15(2): 681-701 doi: 10.3934/dcds.2006.15.681
We investigate the local stability of traveling-wave solutions of the nonlinear reaction-diffusion equations in various weighted Banach spaces. New methods are used in analyzing the location of the spectrum. The result covers the stability results of the traveling-wave solutions of reaction-diffusion equations including the well known Fisher-KPP-type nonlinearity.
keywords: Nonlinear Fisher-KPP-type nonlinearity. Banach spaces reaction diffusion
DCDS
Global existence of solutions to a cross-diffusion system in higher dimensional domains
Yi Li Chunshan Zhao
Discrete & Continuous Dynamical Systems - A 2005, 12(2): 185-192 doi: 10.3934/dcds.2005.12.185
We consider a strongly coupled nonlinear parabolic system which arises from population dynamics in $N$-dimensional $(N\geq 1)$ domains. We establish global existence of classical solutions under certain restrictions on diffusion coefficients, self-diffusion coefficients and cross-diffusion coefficients for both species.
keywords: nonlinear parabolic system population dynamics self-diffusion Shigesada-Kawasaki-Teramoto model. Global existence cross-diffusion
DCDS-B
Stability of travelling waves with noncritical speeds for double degenerate Fisher-Type equations
Yi Li Yaping Wu
Discrete & Continuous Dynamical Systems - B 2008, 10(1): 149-170 doi: 10.3934/dcdsb.2008.10.149
This paper is concerned with the asymptotic stability of travel- ling wave solutions for double degenerate Fisher-type equations. By spectral analysis, each travelling front solution with non-critical speed is proved to be linearly exponentially stable in some exponentially weighted spaces. Further by Evans function method and detailed semigroup estimates each travelling front solution with non-critical speed is proved to be locally algebraically stable to perturbations in some polynomially weighted spaces, and it is also locally exponentially stable to perturbations in some polynomially and exponentially weighted spaces.
keywords: asymptotic stability semigroup estimates travelling waves algebraic decay Evans function. spectral analysis
DCDS-B
Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift
Joaquin Riviera Yi Li
Discrete & Continuous Dynamical Systems - B 2010, 13(1): 157-174 doi: 10.3934/dcdsb.2010.13.157
In this paper we discuss the existence of traveling wave solutions for a nonlocal reaction-diffusion model of Influenza A proposed in Lin et. al. (2003). The proof for the existence of the traveling wave takes advantage of the different time scales between the evolution of the disease and the progress of the disease in the population. Under this framework we are able to use the techniques from geometric singular perturbation theory to prove the existence of the traveling wave.
keywords: multifractal analysis. Poincaré recurrences Dimension theory
CPAA
Traveling waves in a three species competition-cooperation system
Xiaojie Hou Yi Li
Communications on Pure & Applied Analysis 2017, 16(4): 1103-1120 doi: 10.3934/cpaa.2017053

This paper studies the traveling wave solutions to a three species competition cooperation system, which is derived from a spatially averaged and temporally delayed Lotka Volterra system. The existence of the traveling waves is investigated via a new type of monotone iteration method. The upper and lower solutions come from either the waves of KPP equation or those of certain two species Lotka Volterra system. We also derive the asymptotics and uniqueness of the wave solutions.

keywords: Traveling wave spatio-temporal delay Lotka Volterra competition cooperation existence
DCDS
Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities
Xiaojie Hou Yi Li Kenneth R. Meyer
Discrete & Continuous Dynamical Systems - A 2010, 26(1): 265-290 doi: 10.3934/dcds.2010.26.265
This paper studies the traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. The existence, uniqueness, asymptotics as well as the stability of the wave solutions are investigated. The traveling wave solutions, existed for a continuance of wave speeds, do not approach the equilibria exponentially with speed larger than the critical one. While with the critical speed, the wave solutions approach to one equilibrium exponentially fast and to the other equilibrium algebraically. This is in sharp contrast with the asymptotic behaviors of the wave solutions of the classical KPP and $m-th$ order Fisher equations. A delicate construction of super- and sub-solution shows that the wave solution with critical speed is globally asymptotically stable. A simpler alternative existence proof by LaSalle's Wazewski principle is also provided in the last section.
keywords: Asymptotics Uniqueness Heteroclinc Orbits. Traveling Wave Existence
DCDS
Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics
Yunfeng Jia Yi Li Jianhua Wu
Discrete & Continuous Dynamical Systems - A 2017, 37(9): 4785-4813 doi: 10.3934/dcds.2017206

In this paper, a reaction-diffusion system known as an autocatalytic reaction model is considered. The model is characterized by a system of two differential equations which describe a type of complex biochemical reaction. Firstly, some basic characterizations of steady-state solutions of the model are presented. And then, the stability of positive constant steady-state solution and the non-existence, existence of non-constant positive steady-state solutions are discussed. Meanwhile, the bifurcation solution which emanates from positive constant steady-state is investigated, and the global analysis to the system is given in one dimensional case. Finally, a few numerical examples are provided to illustrate some corresponding analytic results.

keywords: Autocatalytic reaction model positive steady-state stability existence and non-existence bifurcation numerical simulation
DCDS
Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus
Ruofei Yao Yi Li Hongbin Chen
Discrete & Continuous Dynamical Systems - A 2018, 0(0): 1-10 doi: 10.3934/dcds.2018122
In this paper, we show the following equation
$\begin{cases} Δ u+u^{p}+λ u = 0&\text{ in }Ω,\\ u = 0&\text{ on }\partialΩ, \end{cases}$
has at most one positive radial solution for a certain range of
$λ>0$
. Here
$p>1$
and
$Ω$
is the annulus
$\{x∈{{\mathbb{R}}^{n}}:a<|x|<b\}$
,
$0<a<b$
. We also show this solution is radially non-degenerate via the bifurcation methods.
keywords: Uniqueness bifurcation methods positive radial solution

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