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CPAA publishes original research papers of the highest quality in all the major areas of analysis and its applications, with a central theme on theoretical and numeric differential equations. Invited expository articles are also published from time to time. CPAA is a bimonthly publication, published in January, March, May, July, September and November. It is edited by a group of energetic leaders to guarantee the journal's highest standard and closest link to the scientific communities.
CPAA is issued jointly by the American Institute of Mathematical Sciences and Shanghai Jiao Tong University. All rights reserved.
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TOP 10 Most Read Articles in CPAA, June 2016
1 
Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion
Volume 9, Number 6, Pages: 1617  1637, 2010
Andrea L. Bertozzi
and Dejan Slepcev
Abstract
Full Text
Related Articles
We present an energymethodsbased proof of the existence and uniqueness of solutions of
a nonlocal aggregation equation with degenerate diffusion. The equation we study
is relevant to models of biological aggregation.

2 
Heterogeneityinduced spot dynamics for a threecomponent reactiondiffusion system
Volume 11, Number 1, Pages: 307  338, 2011
Yasumasa Nishiura,
Takashi Teramoto
and Xiaohui Yuan
Abstract
References
Full Text
Related Articles
Spatially localized patterns form a representative class of patterns in dissipative systems.
We study how the dynamics of traveling spots in twodimensional space change when heterogeneities are introduced in the media.
The simplest but fundamental one is a line heterogeneity of jump type. When spots encounter the jump,
they display various outputs including penetration, rebound, and trapping depending on the incident angle and its height.
The system loses translational symmetry by the heterogeneity, but at the same time, it causes the emergence of various
types of heterogeneityinducedorderedpatterns (HIOPs) replacing the homogeneous constant
state. We study these issues by using a threecomponent reactiondiffusion system with one activator
and two inhibitors. The above outputs can be obtained through the
interaction between the HIOPs and the traveling spots.
The global bifurcation and eigenvalue behavior of HISPs are the key to understand the underlying
mechanisms for the transitions among those dynamics. A reduction to a finite dimensional
system is presented here to extract the modelindependent nature
of the dynamics. Selected numerical techniques for the bifurcation analysis are also
provided.

3 
Error analysis of a conservative
finiteelement approximation for the KellerSegel system of chemotaxis
Volume 11, Number 1, Pages: 339  364, 2011
Norikazu Saito
Abstract
References
Full Text
Related Articles
We are concerned with the finiteelement approximation for the KellerSegel system
that describes the aggregation of slime molds resulting from their
chemotactic features.
The scheme makes use of a semiimplicit time discretization
with a timeincrement control and BabaTabata's conservative upwind
finiteelement approximation in order to realize the positivity and
mass conservation properties. The main aim is to present error analysis
that is an application of the discrete version of the analytical semigroup theory.

4 
Bifurcation analysis to RayleighBénard convection in finite box with updown symmetry
Volume 5, Number 2, Pages: 383  393, 2006
Toshiyuki Ogawa
Abstract
Full Text
Related Articles
RayleighBénard convection in a small rectangular domain is studied
by the standard bifurcation analysis.
The dynamics on the center manifold is calculated up to
3rd order. By this normal form, it is possible to determine the local bifurcation
structures in principle. One can easily imagine that mixed mode solutions
such as hexagonal, patchworkquilt patterns are unstable from the knowledge
of amplitude equation:GinzburgLandau equation. However they are not necessarily
similar to each other in a small rectangular domain.
Several nontrivial stable mixed mode patterns are introduced.

5 
Multiple solutions for a class of nonlinear Neumann eigenvalue problems
Volume 13, Number 4, Pages: 1491  1512, 2014
Leszek Gasiński
and Nikolaos S. Papageorgiou
Abstract
References
Full Text
Related Articles
We consider a parametric nonlinear equation driven by the Neumann $p$Laplacian.
Using variational methods we show that when the parameter
$\lambda > \widehat{\lambda}_1$
(where $\widehat{\lambda}_1$ is the first nonzero eigenvalue of the negative Neumann $p$Laplacian),
then the problem has at least three nontrivial smooth solutions,
two of constant sign
(one positive and one negative)
and the third nodal.
In the semilinear case
(i.e., $p=2$),
using Morse theory and flow invariance argument,
we show that the problem has three nodal solutions.

6 
Wellposedness and scattering for a system of quadratic derivative
nonlinear Schrödinger equations with low regularity initial data
Volume 13, Number 4, Pages: 1563  1591, 2014
Hiroyuki Hirayama
Abstract
References
Full Text
Related Articles
In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear
Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laserplasma interaction.
The local existence of the solution of the system in the Sobolev space $H^s$ for $s > d/2+3$ is proved by M. Colin and T. Colin.
We prove the wellposedness of the system with low regularity initial data.
For some cases, we also prove the wellposedness and the scattering at the scaling critical regularity
by using $U^2$ space and $V^2$ space which are applied to prove
the wellposedness and the scattering for KPII equation at the scaling critical regularity by Hadac, Herr and Koch (2009).

7 
Pullback exponential attractors for evolution processes in Banach spaces: Theoretical results
Volume 12, Number 6, Pages: 3047  3071, 2013
Alexandre Nolasco de Carvalho
and Stefanie Sonner
Abstract
References
Full Text
Related Articles
We construct exponential pullback attractors for time continuous
asymptotically compact evolution processes in Banach spaces and
derive estimates on the fractal dimension of the attractors. We
also discuss the corresponding results for autonomous processes.

8 
Existence and regularity results for the primitive equations in two space dimensions
Volume 3, Number 1, Pages: 115  131, 2004
M. Petcu,
Roger Temam
and D. Wirosoetisno
Abstract
Full Text
Related Articles
Our aim in this article is to present some existence, uniqueness and
regularity results for the Primitive Equations of the ocean in space
dimension two with periodic boundary conditions.
We prove the existence of weak solutions for the PEs, the existence and
uniqueness of strong solutions and the existence of more regular solutions,
up to $\mathcal C^\infty$ regularity.

9 
Global existence of solutions for the thermoelastic
Bresse system
Volume 13, Number 4, Pages: 1395  1406, 2014
Yuming Qin,
Xinguang Yang
and Zhiyong Ma
Abstract
References
Full Text
Related Articles
In this paper, using the semigroup approach, we obtain the global existence of solutions for
linear (nonlinear) homogeneous (nonhomogeneous) thermoelastic Bresse System.

10 
Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part
Volume 12, Number 2, Pages: 771  783, 2012
Minbo Yang
and Yanheng Ding
Abstract
References
Full Text
Related Articles
In the present paper we study the existence of solutions for a nonlocal Schrödinger equation
\begin{eqnarray*}
\varepsilon^2\Delta u +V(x)u =(\int_{R^3}
\frac{u^{p}}{xy^{\mu}}dy)u^{p2}u,
\end{eqnarray*}
where $0 < \mu < 3$ and $\frac{6\mu}{3} < p < {6\mu}$. Under suitable assumptions on the potential $V(x)$, if the parameter $\varepsilon$ is small enough, we prove the existence of solutions by using MountainPass Theorem.

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