摘要：
This paper investigates higher order wave-type equations of the form partial derivative(u)u + P(D-x)u = 0, where the symbol P(xi) is a real, non-degenerate elliptic polynomial of the order m >= 4 on R-n. Using methods from harmonic analysis, we first establish global pointwise time-space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the Cauchy problem of such wave equations. These estimates are then used to establish (pointwise-in-time) L-p-L-q estimates on the wave solution M terms of the initial conditions. (C) 2012 Elsevier Inc. All rights reserved.
期刊：
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS,2012年14(6):1250044 ISSN：0219-1997
通讯作者：
Wang, Chunhua
作者机构：
[Wang, Chunhua] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Wang, Jiangtao] Zhongnan Univ Econ & Law, Sch Math & Stat, Wuhan 430073, Peoples R China.
通讯机构：
[Wang, Chunhua] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Hardy–Sobolev–Maz'ya equation;infinitely many solutions;variational methods
摘要：
In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy–Sobolev–Maz'ya equation with critical growth: provided N > 6 + t, where when k > 2, λ = 0 when k = 2, and Ω is an open bounded domain in ℝN, which contains some points x0 = (0, z0).
期刊：
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES,2012年22(7):1250012 ISSN：0218-2025
通讯作者：
Duan, Renjun
作者机构：
[Duan, Renjun] Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China.;[Ruan, Lizhi; Zhu, Changjiang] Cent China Normal Univ, Hubei Key Lab Math Phys, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Duan, Renjun] C;Chinese Univ Hong Kong, Dept Math, Shatin, Hong Kong, Peoples R China.
关键词：
Scalar conservation laws;weak dissipation;time-weighted energy method;optimal time-decay rate;large-time behavior
摘要：
We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s ∈ ℝ over the whole space ℝn for any spatial dimension n ≥ 1. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for s < 1 and s > 1, respectively. For all s ∈ ℝ, we not only obtain the Lp–Lq time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.
作者机构：
[Li, Shuchao; Yang, Huangxu; Zhao, Qin] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Cactus graphs;Pendant vertex;Zagreb indices
摘要：
For a (molecular) graph, the first Zagreb indexM1 is equal to the sum of squares of its vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this paper, we investigate the first and the second Zagreb indices of cacti with k pendant vertices. We determine sharp bounds for M1-, M2-values of n-vertex cacti with k pendant vertices. As a consequence, we determine the n-vertex cacti with maximal Zagreb indices and we also determine the cactus with a perfect matching having maximal Zagreb indices.
摘要：
In this paper, we consider the following problem
$$
\left\{
\begin{array}{ll}
-\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\
u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N},
\end{array}
\right. (\star)
$$
where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We
prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has
exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by
Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover,
if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a
turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
作者机构：
[Deng, Qin-Tao] Huazhong Normal Univ, Lab Nonlinear Anal, Wuhan 430079, Peoples R China.;[Gu, Hui-Ling; Su, Yan-Hui] Sun Yat Sen Univ, Dept Math, Guangzhou 510275, Guangdong, Peoples R China.
通讯机构：
[Deng, Qin-Tao] H;Huazhong Normal Univ, Lab Nonlinear Anal, Wuhan 430079, Peoples R China.
关键词：
53C40;53C42
摘要：
In this paper, we first summarise the progress for the famous Chern conjecture, and then we consider n-dimensional closed hypersurfaces with constant mean curvature H in the unit sphere Sn+1 with n <= 8 and generalise the result of Cheng et al. (Q. M. Cheng, Y. J. He and H. Z. Li, Scalar curvature of hypersurfaces with constant mean curvature in a sphere, Glasg. Math. J. 51(2) (2009), 413-423). In order to be precise, we prove that if vertical bar H vertical bar <= epsilon(n), then there exists a constant delta(n, H) > 0, which depends only on n and H, such that if S-0 <= S <= S-0 + delta(n, H), then S = S-0 and M is isometric to the Clifford hypersurface, where epsilon(n) is a sufficiently small constant depending on n.
作者机构：
[Chen, Guantao] Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.;[Hu, Zhiquan] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Li, Hao] Univ Paris Sud, LRI, UMR CNRS UPS 8623, F-91405 Orsay, France.
通讯机构：
[Chen, Guantao] G;Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.
会议名称：
8th French Combinatorial Conference (FCC)
会议时间：
JUN 28-JUL 02, 2010
会议地点：
Orsay, FRANCE
会议主办单位：
[Chen, Guantao] Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.^[Hu, Zhiquan] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.^[Li, Hao] Univ Paris Sud, LRI, UMR CNRS UPS 8623, F-91405 Orsay, France.
关键词：
Cycle;Path;Degree;Toughness;Connectivity
摘要：
A path in a graph is called extendable if it is a proper subpath of another path. A graph is locally connected if every neighborhood induces a connected subgraph. We show that, for each graph G of order n, there exists a threshold number s such that every path of order smaller than s is extendable and there exists a non-extendable path of order t for each t is an element of {s, ... , n - 1} if G satisfies any one of the following three conditions: the degree sum d(u) + d(nu) >= n for any two nonadjacent vertices u and nu; P-4-free and omega(G - S) <= vertical bar S vertical bar for every cut set S of G: connected, locally connected, and K-1.3-free where P-4 and K-1,K-3 denote a path of order 4 and a complete bipartite graph with 1 and 3 vertices in each color class, respectively. (C) 2011 Elsevier B.V. All rights reserved.
