DCDS

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.

DCDS-B

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.

DCDS

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.

DCDS

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.

DCDS-B

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.

DCDS-B

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.

CPAA

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.

DCDS

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.

DCDS

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.

DCDS

In this paper, we show the following equation

has at most one positive radial solution for a certain range of

. Here

and

is the annulus

,

. We also show this solution is radially non-degenerate via the bifurcation methods.