26. CMB 2011 (vol 56 pp. 378)
 Ma, Li; Wang, Jing

Sharp Threshold of the GrossPitaevskii Equation with Trapped Dipolar Quantum Gases
In this paper, we consider the GrossPitaevskii equation for the
trapped dipolar quantum gases. We obtain the sharp criterion for the
global existence and finite time blow up in the unstable regime by
constructing a variational problem and the socalled invariant
manifold of the evolution flow.
Keywords:GrossPitaevskii equation, sharp threshold, global existence, blow up Categories:35Q55, 35A05, 81Q99 

27. CMB 2011 (vol 56 pp. 659)
 Yu, ZhiXian; Mei, Ming

Asymptotics and Uniqueness of Travelling Waves for NonMonotone Delayed Systems on 2D Lattices
We establish asymptotics and uniqueness (up
to translation) of travelling waves for delayed 2D lattice equations
with nonmonotone birth functions. First, with the help of
Ikehara's Theorem, the a priori asymptotic behavior of
travelling wave is exactly derived. Then, based on the obtained
asymptotic behavior, the uniqueness of the traveling waves is
proved. These results complement earlier results in the literature.
Keywords:2D lattice systems, traveling waves, asymptotic behavior, uniqueness, nonmonotone nonlinearity Category:35K57 

28. CMB 2011 (vol 56 pp. 3)
 Aïssiou, Tayeb

Semiclassical Limits of Eigenfunctions on Flat $n$Dimensional Tori
We provide a proof of a conjecture by Jakobson, Nadirashvili, and
Toth stating
that on an $n$dimensional flat torus $\mathbb T^{n}$, and the Fourier transform
of squares of the eigenfunctions $\varphi_\lambda^2$ of the Laplacian have
uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof
is a generalization of an argument by Jakobson, et al. for the
lower dimensional cases. These results imply uniform bounds for semiclassical
limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of
codimensionone simplices satisfying a certain restriction on an
$n$dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in
the proof.
Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits Categories:58G25, 81Q50, 35P20, 42B05 

29. CMB 2011 (vol 55 pp. 555)
 Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang

Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications
In this paper we prove weighted norm inequalities with weights in
the $A_p$ classes, for pseudodifferential operators with symbols in
the class ${S^{n(\rho 1)}_{\rho, \delta}}$ that fall outside the
scope of CalderÃ³nZygmund theory. This is accomplished by
controlling the sharp function of the pseudodifferential operator by
HardyLittlewood type maximal functions. Our weighted norm
inequalities also yield $L^{p}$ boundedness of commutators of
functions of bounded mean oscillation with a wide class of operators
in $\mathrm{OP}S^{m}_{\rho, \delta}$.
Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates Categories:42B20, 42B25, 35S05, 47G30 

30. CMB 2011 (vol 55 pp. 663)
 Zhou, Chunqin

An Onofritype Inequality on the Sphere with Two Conical Singularities
In this paper, we give a new proof of the Onofritype inequality
\begin{equation*}
\int_S e^{2u} \,ds^2 \leq 4\pi(\beta+1) \exp \biggl\{
\frac{1}{4\pi(\beta+1)} \int_S \nabla u^2 \,ds^2 +
\frac{1}{2\pi(\beta+1)} \int_S u \,ds^2 \biggr\}
\end{equation*}
on the sphere $S$ with Gaussian curvature $1$ and with conical
singularities divisor $\mathcal A = \beta\cdot p_1 + \beta \cdot p_2$ for
$\beta\in (1,0)$; here $p_1$ and $p_2$ are antipodal.
Categories:53C21, 35J61, 53A30 

31. CMB 2011 (vol 55 pp. 736)
32. CMB 2011 (vol 55 pp. 537)
33. CMB 2011 (vol 55 pp. 623)
 Pan, Jiaqing

The Continuous Dependence on the Nonlinearities of Solutions of Fast Diffusion Equations
In this paper, we consider the Cauchy problem
$$
\begin{cases}
u_{t}=\Delta(u^{m}), &x\in{}\mathbb{R}^{N}, t>0, N\geq3,
\\
% ^^ here
u(x,0)=u_{0}(x), &x\in{}\mathbb{R}^{N}.
\end{cases}
$$
We will prove that:
(i) for
$m_{c}
Keywords:fast diffusion equations, Cauchy problem, continuous dependence on nonlinearity Categories:35K05, 35K10, 35K15 

34. CMB 2011 (vol 55 pp. 249)
 Chang, DerChen; Li, Bao Qin

Description of Entire Solutions of Eiconal Type Equations
The paper describes entire solutions to the eiconal type nonlinear partial differential
equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases,
where
$X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$,
$X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$
are linearly independent operators with $p_j$ being arbitrary
polynomials in $\mathbf{C}^2$.
Keywords:entire solution, eiconal equation, polynomial, transcendental function Categories:32A15, 35F20 

35. CMB 2011 (vol 55 pp. 3)
36. CMB 2011 (vol 55 pp. 88)
 Ghanbari, K.; Shekarbeigi, B.

Inequalities for Eigenvalues of a General Clamped Plate Problem
Let $D$ be a
connected bounded domain in $\mathbb{R}^n$. Let
$0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues
of the following Dirichlet
problem:
$$
\begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in
D
u_{\partial D}=\frac{\partial u}{\partial n}_{\partial
D}=0,
\end{cases}
$$
where $V(x)$ is a nonnegative potential,
and $\rho(x)\in C(\bar{D})$ is positive.
We prove the following inequalities:
$$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}}
{\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times
\frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}\mu_i)]^{1/2},
$$
$$\frac{n^2k^2}{8(n+2)}\leq
\Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}\mu_i}\Bigr]
\times\sum_{i=1}^k\mu_i^{1/2}.
$$
Keywords:biharmonic operator, eigenvalue, eigenvector, inequality Category:35P15 

37. CMB 2011 (vol 54 pp. 249)
38. CMB 2010 (vol 54 pp. 28)
 Chang, YuHsien; Hong, ChengHong

Generalized Solution of the Photon Transport Problem
The purpose of this paper is to show the existence of a
generalized solution of the photon transport problem. By means of the theory of
equicontinuous $C_{0}$semigroup on a sequentially complete locally convex
topological vector space we show that the perturbed abstract Cauchy problem
has a unique solution when the perturbation operator and the forcing term
function satisfy certain conditions. A consequence of the abstract result is
that it can be directly applied to obtain a generalized solution of the photon
transport problem.
Keywords:photon transport, $C_{0}$semigroup Categories:35K30, 47D03 

39. CMB 2010 (vol 54 pp. 126)
40. CMB 2010 (vol 53 pp. 674)
 Kristály, Alexandru; Papageorgiou, Nikolaos S.; Varga, Csaba

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary
We study a semilinear elliptic problem on a compact Riemannian
manifold with boundary, subject to an inhomogeneous Neumann
boundary condition. Under various hypotheses on the nonlinear
terms, depending on their behaviour in the origin and infinity, we
prove multiplicity of solutions by using variational arguments.
Keywords:Riemannian manifold with boundary, Neumann problem, sublinearity at infinity, multiple solutions Categories:58J05, 35P30 

41. CMB 2010 (vol 53 pp. 737)
42. CMB 2009 (vol 53 pp. 295)
 Guo, Boling; Huo, Zhaohui

The Global Attractor of a Damped, Forced Hirota Equation in $H^1$
The existence of the global attractor of a damped
forced Hirota equation in the phase space $H^1(\mathbb R)$ is proved. The
main idea is to establish the socalled asymptotic compactness
property of the solution operator by energy equation approach.
Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactness Categories:35Q53, 35B40, 35B41, 37L30 

43. CMB 2009 (vol 53 pp. 163)
44. CMB 2009 (vol 53 pp. 153)
45. CMB 2009 (vol 52 pp. 555)
 Hirata, Kentaro

Boundary Behavior of Solutions of the Helmholtz Equation
This paper is concerned with the boundary behavior of solutions of
the Helmholtz equation in $\mathbb{R}^\di$.
In particular, we give a Littlewoodtype theorem to show that
the approach region introduced by Kor\'anyi and Taylor (1983) is best possible.
Keywords:boundary behavior, Helmholtz equation Categories:31B25, 35J05 

46. CMB 2009 (vol 40 pp. 60)
 Khavinson, Dmitry

Cauchy's problem for harmonic functions with entire data on a sphere
We give an elementary potentialtheoretic proof of a theorem of
G.~Johnsson: all solutions of Cauchy's problems for the Laplace
equations with an entire data on a sphere extend harmonically to
the whole space ${\bf R}^N$ except, perhaps, for the center of the
sphere.
Keywords:harmonic functions, Cauchy's problem, homogeneous harmonics Categories:35B60, 31B20 

47. CMB 2009 (vol 40 pp. 464)
 Kuo, ChungCheng

On the solvability of a Neumann boundary value problem at resonance
We study the existence of solutions of the semilinear equations (1)
$\triangle u + g(x,u)=h$, ${\partial u \over \partial n} = 0$ on
$\partial \Omega$ in which the nonlinearity $g$ may grow
superlinearly in $u$ in one of directions $u \to \infty$ and $u \to
\infty$, and (2) $\triangle u + g(x,u)=h$, ${\partial u \over
\partial n} = 0$ on $\partial \Omega$ in which the nonlinear term $g$
may grow superlinearly in $u$ as $u \to \infty$. The purpose of this
paper is to obtain solvability theorems for (1) and (2) when the
LandesmanLazer condition does not hold. More precisely, we require
that $h$ may satisfy $\int g^\delta_ (x) \, dx < \int h(x) \, dx = 0<
\int g^\gamma_+ (x)\,dx$, where $\gamma, \delta$ are arbitrarily
nonnegative constants, $g^\gamma_+ (x) = \lim_{u \to \infty} \inf
g(x,u) u^\gamma$ and $g^\delta_ (x)=\lim_{u \to \infty} \sup
g(x,u)u^\delta$. The proofs are based upon degree theoretic arguments.
Keywords:LandesmanLazer condition, Leray Schauder degree Categories:35J65, 47H11, 47H15 

48. CMB 2009 (vol 40 pp. 244)
 Naito, Yūki; Usami, Hiroyuki

Nonexistence results of positive entire solutions for quasilinear elliptic inequalities
This paper treats the quasilinear elliptic inequality
$$
\div (Du^{m2}Du) \geq p(x)u^{\sigma},
\quad x \in \Rs^N,
$$
where $N \geq 2$, $m > 1$, $ \sigma > m  1$, and $p \colon \Rs^N
\rightarrow (0, \infty)$ is continuous. Sufficient conditions are
given for this inequality to have no positive entire solutions. When
$p$ has radial symmetry, the existence of positive entire solutions can
be characterized by our results and some known results.
Categories:35J70, 35B05 

49. CMB 2009 (vol 40 pp. 174)
50. CMB 2008 (vol 51 pp. 249)
 Mangoubi, Dan

On the Inner Radius of a Nodal Domain
Let $M$ be a closed Riemannian manifold.
We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$.
We give upper and lower bounds on the inner radius of the type
$C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on
a local behavior of eigenfunctions discovered by Donnelly and
Fefferman and a Poincar\'{e} type inequality proved by Maz'ya.
Sharp lower bounds are known
only in dimension two. We give an account of this case too.
Categories:58J50, 35P15, 35P20 
