## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - B
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DCDS-B

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.

DCDS-B

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.

CPAA

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.

CPAA

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.

AMC

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.

DCDS-B

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.

DCDS

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 $p$.
CPAA

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.

DCDS-B

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.

CPAA

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.

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