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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, May 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 
Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms
Volume 7, Number 1, Pages: 163  192, 2007
Olivier Guibé
and Anna Mercaldo
Abstract
Full Text
Related Articles
In the present paper we prove uniqueness results for weak
solutions to a class of problems whose prototype is
$d i v((1+\nabla u^2)^{(p2)/2} \nabla u)d i v(c(x) (1+u^2)^{(\tau+1)/2}) $
$+b(x) (1+\nabla u^2)^{(\sigma+1)/2}=f \ i n \ \mathcal D'(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
$u\in W^{1,p}_0(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$
where $\Omega$ is a bounded open subset of $\mathbb R^N$ $(N\ge 2)$,
$p$ is a real number $\frac{2N}{N+1}< p <+\infty$, the coefficients
$c(x)$ and $b(x)$ belong to suitable Lebesgue spaces,
$f$ is an element of the dual space $W^{1,p'}(\Omega)$ and $\tau$ and $\sigma$
are positive constants which belong to suitable intervals specified
in Theorem 2.1, Theorem 2.2 and Theorem 2.3.

3 
Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains
Volume 8, Number 1, Pages: 295  309, 2008
Giuseppe Geymonat
and Françoise Krasucki
Abstract
Full Text
Related Articles
In 1999 M. Eastwood has used the general construction known as the BernsteinGelfandGelfand (BGG) resolution to prove, at least in smooth situation, the equivalence of the linear elasticity complex and of the de Rham complex in $\mathbf{R}^{3}$. The main objective of this paper is to study the linear elasticity complex for general Lipschitz domains in $\mathbf{R}^{3}$ and deduce a complete Hodge orthogonal decomposition for symmetric matrix fields in $L^{2}$, counterpart of the Hodge decomposition for vector fields. As a byproduct one obtains that the finite dimensional terms of this Hodge decomposition can be interpreted in homological terms as the corresponding terms for the de Rham complex if one takes the homology with value in $RIG\cong \mathbf{R}^{6}$ as in the (BGG) resolution.

4 
Traveling waves and their stability in a coupled reaction diffusion system
Volume 10, Number 1, Pages: 141  160, 2010
Xiaojie Hou
and Wei Feng
Abstract
References
Full Text
Related Articles
We study the traveling wave solutions to a reaction diffusion system
modeling the public goods game with altruistic behaviors. The
existence of the waves is derived through monotone iteration of a
pair of classical upper and lower solutions. The waves are shown to
be unique and strictly monotonic. A similar KPP wave like asymptotic
behaviors are obtained by comparison principle and exponential
dichotomy. The stability of the traveling waves with noncritical
speed is investigated by spectral analysis in the weighted Banach
spaces.

5 
Three nontrivial solutions for periodic problems with the $p$Laplacian and a $p$superlinear nonlinearity
Volume 8, Number 4, Pages: 1421  1437, 2009
Leszek Gasiński
and Nikolaos S. Papageorgiou
Abstract
Full Text
Related Articles
We consider a nonlinear periodic problem driven by the scalar
$p$Laplacian and a nonlinearity that exhibits a $p$superlinear growth
near $\pm\infty$, but need not satisfy the AmbrosettiRabinowitz
condition.
Using minimax methods, truncations techniques and Morse theory,
we show that the problem has at least three nontrivial solutions,
two of which are of fixed sign.

6 
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.

7 
Robust exponential attractors for nonautonomous equations with memory
Volume 10, Number 3, Pages: 885  915, 2010
Peter E. Kloeden,
José Real
and Chunyou Sun
Abstract
References
Full Text
Related Articles
The aim of this paper is to consider the robustness of exponential
attractors for nonautonomous dynamical systems with line memory
which is expressed through convolution integrals. Some properties
useful for dealing with the memory term for nonautonomous case
are presented. Then, we illustrate the abstract results by
applying them to the nonautonomous strongly damped wave equations
with linear memory and critical nonlinearity.

8 
Nonexistence for $p$Laplace equations with singular weights
Volume 9, Number 5, Pages: 1421  1438, 2010
Patrizia Pucci
and Raffaella Servadei
Abstract
Full Text
Related Articles
Aim of this paper is to give some nonexistence results of
nontrivial solutions for the following quasilinear elliptic
equations with singular weights in $R^n\setminus \{0\}$
$ \Delta_p u+\mux^{\alpha} u^{a2}u+\lambda 
u^{q2}u+h(x)f(u) = 0 $ and
$ \Delta_p u+\mux^{\alpha} u^{p^*_\alpha2}u+\lambda  u^{q2}u+h(x)f(u)= 0, $
where $1 < p < n$, $\alpha \in [0,p]$, $a \in [p,p^*_\alpha]$,
$p_\alpha^*= p(n\alpha)/(np)$, $\lambda, \mu \in R$ and
$q \ge 1$, while $h: R^+ \to R^+_0$ and $f: R\to R$ are given
continuous functions. The main tool for deriving nonexistence
theorems for the equations is a Pohozaevtype identity. We first
show that such identity holds true for weak solutions $u$ in
$H^{1,p}(R^n)\cap C^1(R^n \setminus \{0\})$ of the first equation
and for weak solutions $u$ in $D^{1,p}(R^n)\cap
C^1(R^n \setminus \{0\})$ of the second equation. Then, under a
suitable growth condition on $f$, we prove that every weak solution
$u$ has the required regularity, so that the Pohozaevtype identity
can be applied. From this identity we derive some nonexistence
results, improving several theorems already appeared in the
literature. In particular, we discuss the case when $h$ and $f$ are
pure powers.

9 
The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains
Volume 7, Number 6, Pages: 1295  1333, 2008
Dorina Mitrea,
Marius Mitrea
and Sylvie Monniaux
Abstract
Full Text
Related Articles
We formulate and solve the Poisson problem for the exterior
derivative operator with Dirichlet boundary condition in
Lipschitz domains, of arbitrary topology, for data in
Besov and TriebelLizorkin spaces.

10 
Bifurcations of some elliptic problems with a singular
nonlinearity via Morse index
Volume 10, Number 2, Pages: 507  525, 2010
Zongming Guo,
Zhongyuan Liu,
Juncheng Wei
and Feng Zhou
Abstract
References
Full Text
Related Articles
We study the boundary value problem
$\Delta u=\lambda x^\alpha f(u)$ in $\Omega, u=1$ on $\partial \Omega\qquad$ (1)
where $\lambda>0$, $\alpha \geq 0$, $\Omega$ is a bounded smooth
domain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ function
satisfying
$\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$,
there is a critical
power $p_c (\alpha)>0$, which is
decreasing in $\alpha$, such that the branch of positive
solutions possesses infinitely many bifurcation
points provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies on
the shape of the domain $\Omega$. We get some important estimates
of the Morse index of the regular and
singular solutions. Moreover, we also study the radial solution
branch of the related problems in the unit ball. We find that the
branch possesses infinitely many turning points provided that
$p>p_c (\alpha)$ and the Morse index of any radial solution (regular or
singular) in this branch is finite provided that
$0 < p \leq p_c (\alpha)$. This implies that the structure of the
radial solution branch of (1) changes for $0 < p \leq p_c
(\alpha)$ and $p > p_c (\alpha)$.

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