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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, July 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
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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
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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 
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
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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.

5 
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.

6 
Global existence of strong solutions to incompressible MHD
Volume 13, Number 4, Pages: 1553  1561, 2014
Huajun Gong
and Jinkai Li
Abstract
References
Full Text
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We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of $\mathbb R^3$ under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $\\sqrt\rho_0u_0\_{L^2(\Omega)}^2+\H_0\_{L^2(\Omega)}^2$ and $\\nabla u_0\_{L^2(\Omega)}^2+\\nabla H_0\_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.

7 
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
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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).

8 
Infiniteenergy solutions for the NavierStokes equations in a strip revisited
Volume 13, Number 4, Pages: 1361  1393, 2014
Peter Anthony
and Sergey Zelik
Abstract
References
Full Text
Related Articles
The paper deals with the NavierStokes equations in a strip in
the class of spatially nondecaying (inniteenergy) solutions belonging to the
properly chosen uniformly local Sobolev spaces. The global wellposedness
and dissipativity of the NavierStokes equations in a strip in such spaces has
been rst established in [22]. However, the proof given there contains a rather
essential error and the aim of the present paper is to correct this error and to
show that the main results of [22] remain true.

9 
Semi discrete weakly damped nonlinear KleinGordon Schrödinger system
Volume 13, Number 4, Pages: 1525  1539, 2014
Olivier Goubet
and Marilena N. Poulou
Abstract
References
Full Text
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We consider a semidiscrete in time relaxation scheme to
discretize a damped forced nonlinear KleinGordon Schrödinger system.
This provides us with a discrete infinitedimensional dynamical
system. We prove the existence of a finite dimensional global
attractor for this dynamical system.

10 
Low regularity wellposedness for the 2D MaxwellKleinGordon equation in the Coulomb gauge
Volume 13, Number 4, Pages: 1669  1683, 2014
Magdalena Czubak
and Nina Pikula
Abstract
References
Full Text
Related Articles
We consider the MaxwellKleinGordon equation in 2D in the Coulomb gauge. We establish local wellposedness for $s=\frac 14+\epsilon$ for data for the spatial part of the gauge potentials and for $s=\frac 58+\epsilon$ for the solution $\phi$ of the gauged KleinGordon equation. The main tool for handling the wave equations is the product estimate established by D'Ancona, Foschi, and Selberg. Due to low regularity, we are unable to use the conventional approaches to handle the elliptic variable $A_0$, so we provide a new approach.

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