## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
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- Electronic Research Announcements
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- AIMS Mathematics

DCDS-B

This paper is concerned with the instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection

$\frac{\partial u}{\partial t}=\text{div}\left( {{\left| \nabla {{u}^{m}} \right|}^{p-2}}\nabla {{u}^{m}} \right)|-\overrightarrow{\beta }\left( x \right)\cdot \triangledown {{u}^{q}},\ \ \ \ x\in {{\mathbb{R}}^{N}},t>0$ |

where

and

is a vector field defined on

. Here, the orientation of the convection is specified to that with counteracting diffusion, that is

,

. Sufficient conditions are established for the instantaneous shrinking property of solutions with decayed initial datum of supports. For a certain class of initial datum, it is shown that there exists a critical time

such that the supports of solutions are unbounded above for any

, whilst the opposite is the case for any

. In addition, we prove that once the supports of solutions shrink instantaneously, the solutions will vanish in finite time.

$p>1, m,q>0, N≥1$ |

$\overrightarrow{β}(x)$ |

$\mathbb{R}^{N}$ |

$\overrightarrow{β}(x)·(-x)≥0$ |

$x∈\mathbb{R}^N$ |

$τ^*>0$ |

$t < τ^*$ |

$t>τ^*$ |

CPAA

This paper is concerned with the un-stirred chemostat with a toxin-producing
competitor. The novelties of the modified model are the periodicity appearing
in the boundary conditions, the different diffusive coefficients of the nutrient
and the microorganisms, and some kinds of death rates. Both uniform persistence
and global extinction of the microorganisms are established under suitable
conditions in terms of principal eigenvalues of scalar periodic parabolic
eigenvalue problems. Our result implies that the toxin inhibits the sensitive
microorganism indeed. The techniques includes the theories of asymptotic periodic
semi-flows, uniform persistence and the perturbation of global attractor.

keywords:
extinction
,
global attractivity
,
uniform persistence
,
Poincaré map
,
Periodic-parabolic problem

DCDS-B

This paper is concerned with the existence of nontrivial and nonnegative periodic
solutions of a doubly degenerate and singular parabolic equation in non-divergent
form with nonlinear sources. We will determine exact classification for the
exponent values of the source, and so, for the nonexistence of nontrivial periodic
solutions, as well as the existence of those solutions with compact support, and
the existence of positive periodic solutions.

DCDS-B

We discuss the possible existence of uncountable smooth traveling wavefronts
of a degenerate and singular parabolic equation in non-divergence form

$\frac{\partial u}{\partial t} =u^m $div$(|\nabla u|^{p-2}\nabla u)+u^qf(u),$

where $f(s)$ is a positive source taking logistic type as an example. A very interesting phenomenon is the presence of critical values $m_c$ and $q_c$ of the exponent $m$ and $q$. Precisely speaking, only for the case $m$<$m_c$ with $q\ge q_c$ can the family of smooth traveling wavefronts have minimal wave speed. We also discuss the regularity of smooth traveling wavefronts.

DCDS

This paper is concerned with large time behavior of solutions for
the Cauchy problem of a semilinear pseudo-parabolic equation with
small perturbation. It is revealed that small perturbation may
develop large variation of solutions with the evolution of time,
which is similar to that for the standard heat equation with
nonlinear sources.

keywords:
small perturbation
,
blow-up
,
Pseudo-parabolic equation
,
global existence.
,
large time behavior

DCDS-B

We discuss the $\omega$-limit set for the Cauchy problem of the porous medium
equation with initial data in some weighted spaces. Exactly, we show that there
exists some relationship between the $\omega$-limit set of the rescaled initial
data and the $\omega$-limit set of the spatially rescaled version of solutions.
We also give some applications of such a relationship.

CPAA

In this paper, we consider an initial-boundary problem for a fourth-order
nonlinear parabolic
equations. The problem as a model shares the scaling properties of the thin film
equation, or as a model
arises in epitaxial growth of nanoscale thin films.
Our approach lies in the
combination of the energy techniques with some methods based on the framework of
Campanato spaces. Based on the
uniform estimates for the approximate solutions,
we establish the existence of weak solutions.

CPAA

This paper is concerned with the existence of positive periodic solutions to
a nonlinear fourth-order differential equation. By virtue of the first
positive eigenvalue of the linear equation corresponding to the nonlinear
fourth order equation, we establish the existence result by using the fixed
point index theory in a cone.

DCDS-B

This paper concerns a convective nonlinear diffusion equation
which is strongly degenerate. The existence and uniqueness
of the $BV$ solution to the initial-boundary problem are proved.
Then we deal with the anti-shifting phenomenon
by investigating the corresponding
free boundary problem. As a consequence, it is possible to find a
suitable convection such that the discontinuous point of the
solution remains unmoved.

CPAA

We discuss non-sharp travelling waves of a dual porous medium equation with monostable source and bistable source respectively.
We show the existence of non-sharp
travelling waves and find that though the equation is degenerate, the travelling waves are classical ones.
Furthermore, for the monostable source, we show that the non-sharp travelling waves are infinite,
while for the bistable source, the non-sharp travelling waves are semi-finite, which is in contrast with the case of the heat equation.

## Year of publication

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