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26. CMB 2011 (vol 56 pp. 378)

Ma, Li; Wang, Jing
Sharp Threshold of the Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases
In this paper, we consider the Gross-Pitaevskii 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 so-called invariant manifold of the evolution flow.

Keywords:Gross-Pitaevskii equation, sharp threshold, global existence, blow up
Categories:35Q55, 35A05, 81Q99

27. CMB 2011 (vol 56 pp. 659)

Yu, Zhi-Xian; Mei, Ming
Asymptotics and Uniqueness of Travelling Waves for Non-Monotone Delayed Systems on 2D Lattices
We establish asymptotics and uniqueness (up to translation) of travelling waves for delayed 2D lattice equations with non-monotone 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

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 codimension-one 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 Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood 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 Onofri-type Inequality on the Sphere with Two Conical Singularities
In this paper, we give a new proof of the Onofri-type 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)

Hernández, Eduardo; O'Regan, Donal
Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations
In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

Keywords:neutral equations, mild solutions, classical solutions
Categories:35R10, 34K40, 34K30

32. CMB 2011 (vol 55 pp. 537)

Kang, Dongsheng
Asymptotic Properties of Solutions to Semilinear Equations Involving Multiple Critical Exponents
In this paper, we investigate a semilinear elliptic equation that involves multiple Hardy-type terms and critical Hardy-Sobolev exponents. By the Moser iteration method and analytic techniques, the asymptotic properties of its nontrivial solutions at the singular points are investigated.

Keywords:elliptic problem, solution, Hardy-Sobolev inequality, singularity, Moser iteration
Categories:35B33, 35B40, 35J60

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, Der-Chen; Li, Bao Qin
Description of Entire Solutions of Eiconal Type Equations
The paper describes entire solutions to the eiconal type non-linear 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)

Agarwal, Ravi P.; Mustafa, Octavian G.
On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations
We improve several recent results in the asymptotic integration theory of nonlinear ordinary differential equations via a variant of the method devised by J. K. Hale and N. Onuchic The results are used for investigating the existence of positive solutions to certain reaction-diffusion equations.

Keywords:asymptotic integration, Emden-Fowler differential equation, reaction-diffusion equation
Categories:34E10, 34C10, 35Q35

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

37. CMB 2011 (vol 54 pp. 249)

Dattori da Silva, Paulo L.
A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies
This paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic.

Keywords:semi-global solvability, analytic solvability, normalization, complex vector fields, condition~($\mathcal P$)
Categories:35A01, 58Jxx

38. CMB 2010 (vol 54 pp. 28)

Chang, Yu-Hsien; Hong, Cheng-Hong
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)

Jin, Yongyang; Zhang, Genkai
Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups
We prove that the fundamental solutions of Kohn sub-Laplacians $\Delta + i\alpha \partial_t$ on the anisotropic Heisenberg groups are tempered distributions and have meromorphic continuation in $\alpha$ with simple poles. We compute the residues and find the partial fundamental solutions at the poles. We also find formulas for the fundamental solutions for some matrix-valued Kohn type sub-Laplacians on H-type groups.

Categories:22E30, 35R03, 43A80

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)

Vougalter, Vitali
On the Negative Index Theorem for the Linearized Non-Linear Schrödinger Problem
A new and elementary proof is given of the recent result of Cuccagna, Pelinovsky, and Vougalter based on the variational principle for the quadratic form of a self-adjoint operator. It is the negative index theorem for a linearized NLS operator in three dimensions.

Categories:35Q55, 81Q10

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 so-called 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)

Taylor, Michael
Variants of Arnold's Stability Results for 2D Euler Equations
We establish variants of stability estimates in norms somewhat stronger than the $H^1$-norm under Arnold's stability hypotheses on steady solutions to the Euler equations for fluid flow on planar domains.


44. CMB 2009 (vol 53 pp. 153)

Niu, Pengcheng; Ou, Yafei; Han, Junqiang
Several Hardy Type Inequalities with Weights Related to Generalized Greiner Operator
In this paper, we establish several weighted $L^p (1\lt p \lt \infty)$ Hardy type inequalities related to the generalized Greiner operator by improving the method of Kombe. Then the best constants in inequalities are discussed by introducing new polar coordinates.

Keywords:generalized Greiner operator, polar coordinates, Hardy inequality
Categories:35B05, 35H99

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 Littlewood-type 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 potential-theoretic 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, Chung-Cheng
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 non-linearity $g$ 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 Landesman-Lazer 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:Landesman-Lazer 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 $$ \div (|Du|^{m-2}Du) \geq p(x)u^{\sigma}, \quad x \in \Rs^N, $$ 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)

Hungerbühler, Norbert
Non-uniqueness for the $p$-harmonic flow
If $f_0\colon\Omega\subset \R^m\to S^n$ is a weakly $p$-harmonic map from a bounded smooth domain $\Omega$ in $\R^m$ (with $2
Categories:35K40, 35K55, 35K65

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
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