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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, January 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
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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 
Large time behavior of solution for the full compressible navierstokesmaxwell system
Volume 14, Number 6, Pages: 2283  2313, 2015
Weike Wang
and Xin Xu
Abstract
References
Full Text
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In this paper, the Cauchy problem for the compressible NavierStokesMaxwell equation is studied in $R^3$, the $L^p$ time decay rate for the global smooth solution is established. Our method is mainly based on a detailed analysis to the Green's function of the linearized system and some elaborate energy estimates. To give the explicit representation of the Green's function, we use the Helmholtz decomposition by which we can decompose the solution into two parts and give the expression to each part. Our results show a sharp difference between the decay of solution for NavierStokesMaxwell system and that for the NavierStokes equation.

4 
Global existence and blow up of solutions to a class of pseudoparabolic equations with an exponential source
Volume 14, Number 6, Pages: 2465  2485, 2015
Xiaoli Zhu,
Fuyi Li
and Ting Rong
Abstract
References
Full Text
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In this paper, a class of pseudoparabolic equations with an exponential source is concerned. First, by the elliptic regularity theory, we establish the local existence and uniqueness of solutions. Sequently, the global existence and blow up of solutions with lower initial energy is considered via the potential wells method. Finally, we find a sufficient condition under which the solution blows up without any limit of initial energy by constructing a new functional which falls between the energy functional and Nehari functional. During this process, the properties of global solutions are also studied.

5 
Symmetry of solutions to semilinear equations
involving the fractional laplacian
Volume 14, Number 6, Pages: 2393  2409, 2015
Lizhi Zhang
Abstract
References
Full Text
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Let $0<\alpha<2$ be any real number. Let $\Omega\subset\mathbb{R}^n$ be a bounded or an unbounded domain which is convex and symmetric in $x_1$ direction. We investigate the following Dirichlet problem involving the fractional Laplacian:
\begin{equation}
\left\{\begin{array}{ll}
(\Delta)^{\alpha/2} u(x)=f(x,u), & \qquad x\in\Omega, (1)\\
u(x)\equiv0, & \qquad x\notin\Omega.
\end{array}\right.
\end{equation}
Applying a direct method of moving planes for the fractional Laplacian, with the help of several maximum principles for antisymmetric functions, we prove the monotonicity and symmetry of positive solutions in $x_1$ direction as well as nonexistence of positive solutions under various conditions on $f$ and on the solutions $u$. We also extend the results to some more complicated cases.

6 
On Fractional Schrödinger Equations in sobolev spaces
Volume 14, Number 6, Pages: 2265  2282, 2015
Younghun Hong
and Yannick Sire
Abstract
References
Full Text
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Let $\sigma \in (0,1)$ with $\sigma \neq \frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$:
\begin{eqnarray}
i\partial_tu+(\Delta)^\sigma u+\muu^{p1}u=0, u(0)=u_0\in H^s,
\end{eqnarray}
where $(\Delta)^\sigma$ is the Fourier multiplier of symbol $\xi^{2\sigma}$, and $\mu=\pm 1$. This model has been introduced by Laskin in quantum physics [23]. We establish local wellposedness and illposedness in Sobolev spaces for powertype nonlinearities.

7 
Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon
Volume 14, Number 6, Pages: 2411  2429, 2015
Qiuping Lu
and Zhi Ling
Abstract
References
Full Text
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For a general elliptic problem $\triangle{u} = g(u)$ in $R^N$ with $N \ge 3$, we show that all solutions have compact support and there exists a least energy solution, which is radially symmetric and decreases with respect to $x = r$. With this result we study a singularly perturbed elliptic problem $ \epsilon^{2} \triangle{u} + u^{q1}u = \lambda u + f(u)$ in a bounded domain $\Omega$ with $0 < q < 1 $, $\lambda \ge 0$ and $ u \in H^1_0(\Omega) $. For any $y \in \Omega$, we show that there exists a least energy solution $u_{\epsilon}$, which concentrates around this point $y$ as $\epsilon \to 0$. Conversely when $\epsilon $ is small, the boundary of the set $\{ x \in \Omega  u_{\epsilon}(x)>0 \}$ is a free boundary, where $u_{\epsilon}$ is any nonnegative least energy solution.

8 
Nodal solutions for a quasilinear Schrödinger equation with critical
nonlinearity and nonsquare diffusion
Volume 14, Number 6, Pages: 2487  2508, 2015
Yinbin Deng,
Yi Li
and Xiujuan Yan
Abstract
References
Full Text
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This paper is concerned with a type of quasilinear Schrödinger equations of the form
\begin{eqnarray}
\Delta u+V(x)up\Delta(u^{2p})u^{2p2}u=\lambdau^{q2}u+u^{2p2^{*}2}u,
\end{eqnarray}
where $\lambda>0, N\ge3, 4p < q < 2p2^*, 2^*=\frac{2N}{N2}, 1< p < +\infty$. For any given $k \ge 0$, by using a change of variables and Nehari minimization, we obtain a signchanging minimizer with $k$ nodes.

9 
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.

10 
Dynamics of a hostpathogen system on a bounded spatial domain
Volume 14, Number 6, Pages: 2535  2560, 2015
FengBin Wang,
Junping Shi
and Xingfu Zou
Abstract
References
Full Text
Related Articles
We study a hostpathogen system in a bounded spatial habitat where the environment is closed.
Extinction and persistence of the disease are investigated by appealing to theories of monotone dynamical systems and uniform persistence. We also carry out a bifurcation analysis for steady state solutions, and
the results suggest that a backward bifurcation may occur when the parameters in the system are spatially dependent.

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