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1. CMB Online first

Demirbas, Seckin
 Almost Sure Global Well-posedness for Fractional Cubic SchrÃ¶dinger Equation on Torus In a previous paper, we proved that $1$-d periodic fractional SchrÃ¶dinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s\gt \frac{1-\alpha}{2}$ and globally well-posed for $s\gt \frac{10\alpha-1}{12}$. In this paper we define an invariant probability measure $\mu$ on $H^s$ for $s\lt \alpha-\frac{1}{2}$, so that for any $\epsilon\gt 0$ there is a set $\Omega\subset H^s$ such that $\mu(\Omega^c)\lt \epsilon$ and the equation is globally well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in almost sure sense for $\frac{1-\alpha}{2}\lt \alpha-\frac{1}{2}$, i.e. $\alpha\gt \frac{2}{3}$ in almost sure sense. Keywords:NLS, fractional Schrodinger equation, almost sure global wellposednessCategory:35Q55

2. CMB Online first

Duc, Dinh Thanh; Nhan, Nguyen Du Vi; Xuan, Nguyen Tong
 Inequalities for partial derivatives and their applications We present various weighted integral inequalities for partial derivatives acting on products and compositions of functions which are applied to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations. Keywords:inequality for integral, Opial-type inequality, HÃ¶lder's inequality, partial differential operator, partial differential equationCategories:26D10, 35A23

3. CMB Online first

Tang, Xianhua
 Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on RN This paper is concerned with the following elliptic system of Hamiltonian type $\left\{ \begin{array}{ll} -\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ -\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{array} \right.$ where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta+V$, $W(x, s, t)$ is periodic in $x$ and superlinear in $s$ and $t$ at infinity. We develop a direct approach to find ground state solutions of Nehari-Pankov type for the above system. Especially, our method is applicable for the case when $W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{|\alpha_iu+\beta_iv|}g_i(x, t)t\mathrm{d}t +\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x, t)t\mathrm{d}t,$ where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne 0$ and $a_j\gt b_j^2$, $g_i(x, t)$ and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every $x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$. Keywords:Hamiltonian elliptic system, superlinear, ground state solutions of Nehari-Pankov type, strongly indefinite functionalsCategories:35J50, 35J55

4. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
 Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse HÃ¶lder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained. Keywords:Musielak-Orlicz-Hardy space, magnetic SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequalityCategories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

5. CMB 2013 (vol 56 pp. 827)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
 Erratum to Quantum Limits of Eisenstein Series and Scattering States'' This paper provides an erratum to Y. N. Petridis, N. Raulf, and M. S. Risager, Quantum Limits of Eisenstein Series and Scattering States.'' Canad. Math. Bull., published online 2012-02-03, http://dx.doi.org/10.4153/CMB-2011-200-2. Keywords:quantum limits, Eisenstein series, scattering polesCategories:11F72, 8G25, 35P25

6. CMB 2012 (vol 56 pp. 814)

Petridis, Yiannis N.; Raulf, Nicole; Risager, Morten S.
 Quantum Limits of Eisenstein Series and Scattering States We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak. Keywords:quantum limits, Eisenstein series, scattering polesCategories:11F72, 58G25, 35P25

7. 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 upCategories:35Q55, 35A05, 81Q99

8. 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 nonlinearityCategory:35K57

9. 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 limitsCategories:58G25, 81Q50, 35P20, 42B05

10. 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 CalderÃ³n-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 estimatesCategories:42B20, 42B25, 35S05, 47G30

11. 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

12. 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 solutionsCategories:35R10, 34K40, 34K30

13. 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 iterationCategories:35B33, 35B40, 35J60

14. 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 $m_{c} Keywords:fast diffusion equations, Cauchy problem, continuous dependence on nonlinearityCategories:35K05, 35K10, 35K15 15. CMB 2011 (vol 55 pp. 249) Chang, Der-Chen; Li, Bao Qin  Description of Entire Solutions of Eiconal Type Equations The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations$(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$. Keywords:entire solution, eiconal equation, polynomial, transcendental functionCategories:32A15, 35F20 16. CMB 2011 (vol 55 pp. 3) Agarwal, Ravi P.; Mustafa, Octavian G.  On a Local Theory of Asymptotic Integration for Nonlinear Differential Equations We improve several recent results in the asymptotic integration theory of nonlinear ordinary differential equations via a variant of the method devised by J. K. Hale and N. Onuchic The results are used for investigating the existence of positive solutions to certain reaction-diffusion equations. Keywords:asymptotic integration, Emden-Fowler differential equation, reaction-diffusion equationCategories:34E10, 34C10, 35Q35 17. CMB 2011 (vol 55 pp. 88) Ghanbari, K.; Shekarbeigi, B.  Inequalities for Eigenvalues of a General Clamped Plate Problem Let$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, inequalityCategory:35P15 18. 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 19. 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,$C_{0}$-semigroupCategories:35K30, 47D03 20. 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 21. 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 solutionsCategories:58J05, 35P30 22. CMB 2010 (vol 53 pp. 737) Vougalter, Vitali  On the Negative Index Theorem for the Linearized Non-Linear SchrÃ¶dinger 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 23. 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 compactnessCategories:35Q53, 35B40, 35B41, 37L30 24. 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 inequalityCategories:35B05, 35H99 25. 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. Category:35Q35
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