DCDS-B
Instantaneous shrinking and extinction for a non-Newtonian polytropic filtration equation with orientated convection
Hailong Ye Jingxue Yin
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1743-1755 doi: 10.3934/dcdsb.2017083
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
$p>1, m,q>0, N≥1$
and
$\overrightarrow{β}(x)$
is a vector field defined on
$\mathbb{R}^{N}$
. Here, the orientation of the convection is specified to that with counteracting diffusion, that is
$\overrightarrow{β}(x)·(-x)≥0$
,
$x∈\mathbb{R}^N$
. 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
$τ^*>0$
such that the supports of solutions are unbounded above for any
$t < τ^*$
, whilst the opposite is the case for any
$t>τ^*$
. In addition, we prove that once the supports of solutions shrink instantaneously, the solutions will vanish in finite time.
keywords: Non-Newtonian polytropic filtration orientated convection instantaneous shrinking extinction
CPAA
Global dynamics of the periodic un-stirred chemostat with a toxin-producing competitor
Yifu Wang Jingxue Yin
Communications on Pure & Applied Analysis 2010, 9(6): 1639-1651 doi: 10.3934/cpaa.2010.9.1639
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
Periodic solutions of a non-divergent diffusion equation with nonlinear sources
Chunhua Jin Jingxue Yin
Discrete & Continuous Dynamical Systems - B 2012, 17(1): 101-126 doi: 10.3934/dcdsb.2012.17.101
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.
keywords: Periodic Solution Diffusion Equation Non-Divergence.
DCDS-B
Critical exponents and traveling wavefronts of a degenerate-singular parabolic equation in non-divergence form
Jingxue Yin Chunhua Jin
Discrete & Continuous Dynamical Systems - B 2010, 13(1): 213-227 doi: 10.3934/dcdsb.2010.13.213
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.

keywords: Non-Divergence Traveling Wavefronts Critical Exponents.
DCDS
Small perturbation of a semilinear pseudo-parabolic equation
Yang Cao Jingxue Yin
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 631-642 doi: 10.3934/dcds.2016.36.631
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
MBE
Early and late stage profiles for a chemotaxis model with density-dependent jump probability
Tianyuan Xu Shanming Ji Chunhua Jin Ming Mei Jingxue Yin
Mathematical Biosciences & Engineering 2018, 15(6): 1345-1385 doi: 10.3934/mbe.2018062

In this paper, we derive a chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, this model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole domain, we first estimate the expanding speed of tumour region as $O(t^{β})$ for $ 0 < β < \frac{1}{2}$. Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c>0$ is also obtained.

keywords: Chemotaxis model degenerate diffusion density-dependent jump probability finite speed propagation tumour invasions models porous media diffusion
DCDS-B
Anti-shifting phenomenon of a convective nonlinear diffusion equation
Chunpeng Wang Jingxue Yin Bibo Lu
Discrete & Continuous Dynamical Systems - B 2010, 14(3): 1211-1236 doi: 10.3934/dcdsb.2010.14.1211
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.
keywords: anti-shifting. convective strong degeneracy $BV$ solution
CPAA
Non-sharp travelling waves for a dual porous medium equation
Jing Li Yifu Wang Jingxue Yin
Communications on Pure & Applied Analysis 2016, 15(2): 623-636 doi: 10.3934/cpaa.2016.15.623
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.
keywords: Travelling wave dual porous medium equation.
DCDS-B
$\omega$-limit sets for porous medium equation with initial data in some weighted spaces
Liangwei Wang Jingxue Yin Chunhua Jin
Discrete & Continuous Dynamical Systems - B 2013, 18(1): 223-236 doi: 10.3934/dcdsb.2013.18.223
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.
keywords: weighted space. $\omega$-limit set Porous medium equation
CPAA
Existence of weak solutions for a generalized thin film equation
Changchun Liu Jingxue Yin Juan Zhou
Communications on Pure & Applied Analysis 2007, 6(2): 465-480 doi: 10.3934/cpaa.2007.6.465
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.
keywords: existence Thin film equation Campanato space.

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