DCDS-B

The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.

DCDS

Existence, uniqueness, and stability of
Heaviside function like solutions
of a Keller and Segel's minimal chemotaxis model
are established when a chemotaxis parameter is large enough.
Asymptotic expansions of the solution
in terms of the large chemotaxis parameter are also derived.

DCDS

This paper is concerned with a modified two-component periodic
Camassa-Holm system. The local well-posedness and low regularity
result of solution are established by using the techniques of
pseudoparabolic regularization and some priori estimates derived
from the equation itself. A wave-breaking for strong solutions and
several results of blow-up solution with certain initial profiles
are described. In addition, the initial boundary value problem for a
modified two-component periodic Camassa-Holm system is also
considered.

DCDS

This paper deals with blow-up properties of the solution to a
semi-linear parabolic system with nonlinear localized sources
involved in a product with local terms, subject to the null
Dirichlet boundary condition. We investigate the influence of
localized sources and local terms on blow-up properties for this
system. It will be proved that: (i) when $m, q\leq 1$ this
system possesses uniform blow-up profiles. In other words, the
localized terms play a leading role in the blow-up profile for
this case. (ii) when $m, q>1$, this system presents single
point blow-up patterns, or say that, in this time, local terms
dominate localized terms in the blow-up profile. Moreover, the
blow-up rate estimates in time and space are obtained,
respectively.

DCDS-B

This paper is concerned with the existence, uniqueness and asymptotically stability of traveling wave fronts of
discrete quasi-linear equations with delay. We first establish the existence of traveling wave fronts by using
the super-sub solution and monotone iteration technique. Then we show that the traveling wave front is unique up
to a translation. At last, we employ the comparison principle and the squeezing technique to prove that the
traveling wave front is globally asymptotic stable with phase shift.

DCDS

In this paper, we study the
variable-territory prey-predator model. We first establish the
global stability of the unique positive constant steady state for
the ODE system and the reaction diffusion system, and then prove
the existence, uniqueness and stability of positive stationary
solutions for the heterogeneous environment case.

DCDS

In this
paper, we focus on the Cauchy problems of nonlinear strongly
damped hyperbolic equations and systems. We give some conditions
on the non-existence of global solutions.

CPAA

In this article we consider the following integral equation involving Bessel potentials on a half space $\mathbb{R}^n_+ $:
\begin{eqnarray}
u(x)=\int_{ \mathbb{R}^n_+ }\{g_\alpha(x-y)-g_\alpha(\bar x-y)\} u^\beta(y) dy,\;\;x\in \mathbb{R}^n_+,
\end{eqnarray}
where $\alpha>0$, $\beta>1$, $\bar
x$ is the reflection of $x$
about $x_n=0$, and $g_\alpha(x)$ denotes the Bessel kernel. We first enhance the regularity of positive solutions for the integral equation by regularity-lifting-method, which has been extensively used by many authors. Then, employing the method of moving planes in integral forms, we demonstrate that there is no positive
solution for the integral equation.

DCDS-B

This paper is devoted to study the dynamical properties of a Leslie-Gower prey-predator system with strong Allee effect in prey. We first gives some estimates, and then study the dynamical properties of solutions. In particular, we mainly investigate the unstable and stable manifolds of the positive equilibrium when the system has only one positive equilibrium.

DCDS

In this paper we investigate a free boundary problem for the diffusive Leslie-Gower prey-predator model with double free boundaries in one space dimension. This system models the expanding of an invasive or new predator species in which the free boundaries represent expanding fronts of the predator species. We first prove the existence, uniqueness and regularity of global solution. Then provide a spreading-vanishing dichotomy, namely the predator species either successfully spreads to infinity as $t\to∞$ at both fronts and survives in the new environment, or it spreads within a bounded area and dies out in the long run. The long time behavior of $(u, v)$ and criteria for spreading and vanishing are also obtained. Because the term $v/u$ (which appears in the second equation) may be unbounded when $u$ nears zero, it will bring some difficulties for our study.