DCDS-B
Allee effect and a catastrophe model of population dynamics
Dianmo Li Zhen Zhang Zufei Ma Baoyu Xie Rui Wang
Some assumptions of Logistic Equation are frequently violated. We applied the Allee effect to the Logistic Equation so as to avoid these unrealistic assumptions. Following basic principles of Catastrophe theory, this new model is identical to a Fold catastrophe type model. An ecological interpretation of the results is provided.
keywords: catastrophe theory Logistic Equation. Allee effect
DCDS-B
Pullback attractors of FitzHugh-Nagumo system on the time-varying domains
Zhen Zhang Jianhua Huang Xueke Pu

The existence and uniqueness of solutions satisfying energy equality is proved for non-autonomous FitzHugh-Nagumo system on a special time-varying domain which is a (possibly non-smooth) domain expanding with time. By constructing a suitable penalty function for the two cases respectively, we establish the existence of a pullback attractor for non-autonomous FitzHugh-Nagumo system on a special time-varying domain.

keywords: Pullback attractor FitzHugh-Nagumo equation time-varying domain penalty function
CPAA
Gradient blowup solutions of a semilinear parabolic equation with exponential source
Zhengce Zhang Yanyan Li
In this paper, we consider the N-dimensional semilinear parabolic equation $ u_t=\Delta u+e^{|\nabla u|}$, for which the spatial derivative of solutions becomes unbounded in finite (or infinite) time while the solutions themselves remain bounded. We establish estimates of blowup rate as well as lower and upper bounds for the radial solutions. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.
keywords: exponential source. Gradient blowup rate estimate
CPAA
Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions
Zhenhua Zhang
This paper is concerned with the asymptotic behavior of solutions to the phase-field equations subject to the Neumann boundary conditions where a Lojasiewicz-Simon type inequality plays an important role. In this paper, convergence of the solution of this problem to an equilibrium, as time goes to infinity, is proved.
keywords: Phase-field equations Lojasiewicz-Simon type inequality gradient system.
AMC
Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties
Zhenyu Zhang Lijia Ge Fanxin Zeng Guixin Xuan
In this paper, a novel method for constructing complementary sequence set with zero correlation zone (ZCZ) is presented by interleaving and combining three orthogonal matrices. The constructed set can be divided into multiple sequence groups and each sequence group can be further divided into multiple sequence subgroups. In addition to ZCZ properties of sequences from the same sequence subgroup, sequences from different sequence groups are orthogonal to each other while sequences from different sequence subgroups within the same sequence group possess ideal cross-correlation properties, that is, the proposed ZCZ sequence set has inter-group orthogonal (IGO) and inter-subgroup complementary (ISC) properties. Compared with previous methods, the new construction can provide flexible choice for ZCZ width and set size, and the resultant sequences which are called IGO-ISC sequences in this paper can achieve the theoretical bound on the set size for the ZCZ width and sequence length.
keywords: inter-group orthogonal sequence Complementary sequence inter-subgroup complementary sequence. zero correlation zone
DCDS-B
Dead-core rates for the heat equation with a spatially dependent strong absorption
Chin-Chin Wu Zhengce Zhang
This work is to study the dead-core behavior for a semilinear heat equation with a spatially dependent strong absorption term. We first give a criterion on the initial data such that the dead-core occurs. Then we prove the temporal dead-core rate is non-self-similar. This is based on the standard limiting process with the uniqueness of the self-similar solutions in a certain class.
keywords: strong absorption. Dead-core heat equation
DCDS
Gradient blowup rate for a semilinear parabolic equation
Zhengce Zhang Bei Hu
We present a one-dimensional semilinear parabolic equation $u_t=$u xx$ +x^m |u_x|^p, p> 0, m\geq 0$, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We show that the spatial derivative of solutions is globally bounded in the case $p\leq m+2$ while blowup occurs at the boundary when $p>m+2$. Blowup rate is also found for some range of $p$.
keywords: Gradient blowup Nonlinear gradient source. Blowup rate
CPAA
An adaptive mesh redistribution algorithm for convection-dominated problems
Zheng-Ru Zhang Tao Tang
Convection-dominated problems are of practical applications and in general may require extremely fine meshes over a small portion of the physical domain. In this work an efficient adaptive mesh redistribution (AMR) algorithm will be developed for solving one- and two-dimensional convection-dominated problems. Several test problems are computed by using the proposed algorithm. The adaptive mesh results are compared with those obtained with uniform meshes to demonstrate the effectiveness and robustness of the proposed algorithm.
keywords: finite volume method Adaptive mesh redistribution convectiondominated.
DCDS-B
Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source
Zhengce Zhang Yan Li
Throughout this paper, we consider the equation \[u_t - \Delta u = e^{|\nabla u|}\] with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\overline{\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.
keywords: stationary solution gradient blowup. Global solution
CPAA
Spike solutions to a nonlocal differential equation
Changfeng Gui Zhenbu Zhang
In this paper we consider a nonlocal differential equation, which is a limiting equation of one dimensional Gierer-Meinhardt model. We study the existence of spike steady states and their stability. We also construct a single-spike quasi-equilibrium solution and investigate the dynamics of spike-like solutions.
keywords: stability. spike solution Gierer-Meinhardt model

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