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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. 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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 Publishes 6 issues a year in January, March, May, July, September and November.
 Publishes both online and in print.
 Indexed in Science Citation Index, CompuMath Citation Index, Current Contents/Physics, Chemical, & Earth Sciences, INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
 Archived in Portico and CLOCKSS.
 CPAA is issued jointly by the American Institute of Mathematical Sciences and Shanghai Jiao Tong University. All rights reserved.

TOP 10 Most Read Articles in CPAA, October 2017
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 
Travelling wave solutions of a free boundary problem for a twospecies competitive model
Volume 12, Number 2, Pages: 1065  1074, 2012
ChuehHsin Chang
and ChiunChuan Chen
Abstract
References
Full Text
Related Articles
We study a diusive logistic system with a free boundary in ecology
proposed by Mimura, Yamada and Yotsutani [10]. Motivated by the spreadingvanishing
dichotomy obtained by Du and Lin [1], we suppose the spreading
speed of the free boundary tends to a constant as time tends to innity and
consider the corresponding travelling wave problem. We establish the existence
and uniqueness of a travelling wave solution for this free boundary problem.

5 
On a general class of free boundary problems for Europeanstyle installment options with continuous payment plan
Volume 10, Number 4, Pages: 1205  1224, 2011
Pierangelo Ciurlia
Abstract
References
Full Text
Related Articles
In this paper we present an integral equation approach for the
valuation of Europeanstyle installment derivatives when the payment
plan is assumed to be a continuous function of the asset price and
time. The contribution of this study is threefold. First, we show
that in the BlackScholes model the option pricing problem can be
formulated as a free boundary problem under very general conditions
on payoff structure and payment schedule. Second, by applying a
Fourier transformbased solution technique, we derive a recursive
integral equation for the free boundary along with an analytic
representation of the option price. Third, based on these results,
we propose a unified framework which generalizes the existing
methods and is capable of dealing with a wide range of monotonic
payoff functions and continuous payment plans. Finally, by using the
illustrative example of European vanilla installment call options,
an explicit pricing formula is obtained for timevarying payment
schedules.

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
Related Articles
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 
Potential well method for initial boundary value problem of the generalized double dispersion equations
Volume 7, Number 1, Pages: 63  81, 2007
Yacheng Liu
and Runzhang Xu
Abstract
Full Text
Related Articles
In this paper we study the initial boundary value problem of the
generalized double dispersion equations
$u_{t t}u_{x x}u_{x x t t}+u_{x x x x}=f(u)_{x x}$, where $f(u)$ include
convex function as a special case. By introducing a family of
potential wells we first prove the invariance of some sets and
vacuum isolating of solutions, then we obtain a threshold result of
global existence and nonexistence of solutions. Finally we discuss
the global existence of solutions for problem with critical initial
condition.

8 
Exterior differential systems and prolongations for
three important nonlinear partial differential equations
Volume 10, Number 5, Pages: 1345  1360, 2011
Paul Bracken
Abstract
References
Full Text
Related Articles
Partial differential systems which have
applications to water waves will be formulated as
exterior differential systems. A prolongation structure
is determined for each of the equations. The formalism
for studying prolongations is reviewed and the
prolongation equations are solved for each equation.
One of these
differential systems includes the CamassaHolm and
DegasperisProcesi equations as special cases.
The formulation of conservation laws for each
of the systems introduced is discussed
and a single example for each is given.
It is shown how a Bäcklund transformation
for the last case can be obtained using
the prolongation results.

9 
Remarks on some dispersive estimates
Volume 10, Number 4, Pages: 1121  1128, 2011
Yonggeun Cho,
Tohru Ozawa
and Suxia Xia
Abstract
References
Full Text
Related Articles
In this paper we consider the initial value problem for $i\partial_t u + \omega(\nabla) u = 0$. Under suitable smoothness and growth conditions on $\omega$, we derive dispersive estimates which is the generalization of time decay and Strichartz estimates. We unify and also simplify dispersive estimates by utilizing the Bessel function. Another main ingredient of this paper is to revisit oscillatory integrals of [2].

10 
An evolution equation involving the normalized $P$Laplacian
Volume 10, Number 1, Pages: 361  396, 2010
Kerstin Does
Abstract
References
Full Text
Related Articles
This paper considers an initialboundary value problem for the evolution equation associated with the normalized $p$Laplacian. There exists a unique viscosity solution $u,$ which is globally Lipschitz continuous with respect to $t$ and locally with respect to $x.$ Moreover, we study the long time behavior of the viscosity solution $u$ and compute numerical solutions of the problem.

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