Blow-up for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations $\{\begin{array}{cc}{u}_{tt}+u_t+{\left|u_t\right|}^{m?1}{u}_t=\mathrm{div}\left({\rho }_1\left({\left|?u\right|}^2\right)?u\right)+f_1(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \\ v_{tt}+v_t+{\left|v_t\right|}^{r?1}{v}_t=\mathrm{div}\left({\rho }_2\left({\left|?v\right|}^2\right)?v\right)+f_2(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \end{array}$ with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions$f_1$, $f_2$, ${\rho }_1$, ${\rho }_2$, parameters $r,m$ and the initial data, the result on blow-up of solutions and upper bound of blow-up time are given.     

Steiner tree packing number and tree connectivity

Let $S$ be a set of at least two vertices in a graph $G$. A subtree $T$ of $G$ is a $S$-Steiner tree if $S?V\left(T\right)$. Two $S$-Steiner trees $T_1$ and $T_2$ are edge-disjoint (resp. internally vertex-disjoint) if $E\left(T_1\right)\cap E\left(T_2\right)=?$ (resp. $E\left(T_1\right)\cap E\left(T_2\right)=?$ and $V\left(T_1\right)\cap V\left(T_2\right)=S$). Let ${\lambda }_G\left(S\right)$ (resp. ${\kappa }_G\left(S\right)$) be the maximum number of edge-disjoint (resp. internally vertex-disjoint) $S$-Steiner trees in $G$, and let ${\lambda }_k\left(G\right)$ (resp. ${\kappa }_k\left(G\right)$) be the minimum ${\lambda }_G\left(S\right)$ (resp. ${\kappa }_G\left(S\right)$) for $S$ ranges over all $k$-subset of $V\left(G\right)$. Kriesell conjectured that if ${\lambda }_G\left(\{x,y\}\right)\ge 2k$ for any $x,y\in S$, then ${\lambda }_G\left(S\right)\ge k$. He proved that the conjecture holds for $\left|S\right|=3,4$. In this paper, we give a short proof of Kriesell’s Conjecture for $\left|S\right|=3,4$, and also show that ${\lambda }_k\left(G\right)\ge ?\frac{1}{k?1}?\frac{k?}{2}??$ (resp. ${\kappa }_k\left(G\right)\ge ?\frac{1}{k?1}?\frac{k?}{2}??$ ) if $\lambda \left(G\right)\ge ?$ (resp. $\kappa \left(G\right)\ge ?$) in $G$, where $k=3,4$. Moreover, we also study the relation between ${\kappa }_k\left(L\left(G\right)\right)$ and ${\lambda }_k\left(G\right)$, where $L\left(G\right)$ is the line graph of $G$.     

Existence and convexity of local solutions to degenerate Hessian equations

In this work, we prove the existence of convex solutions to the following k-Hessian equation

$S_k[u]=K(y)g(y,u,Du)$

in the neighborhood of a point $(y_0,u_0,p_0)\in {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$, where $g\in C^{\infty },g(y_0,u_0,p_0)>0$, $K\in C^{\infty }$ is nonnegative near $y_0$, $K(y_0)=0$ and $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

The local eigenvalues of a bipartite distance-regular graph

We consider a bipartite distance-regular graph $\Gamma$ with vertex set $X$, diameter $D\ge 4$, and valency $k\ge 3$. For $0\le i\le D$, let ${\Gamma }_i\left(x\right)$ denote the set of vertices in $X$ that are distance $i$ from vertex $x$. We assume there exist scalars $r,s,t\in \mathbb{R}$, not all zero, such that $r|{\Gamma }_1\left(x\right)\cap {\Gamma }_1\left(y\right)\cap {\Gamma }_2\left(z\right)|+s|{\Gamma }_2\left(x\right)\cap {\Gamma }_2\left(y\right)\cap {\Gamma }_1\left(z\right)|+t=0$ for all $x,y,z\in X$ with path-length distances $\partial (x,y)=2,\partial (x,z)=3,\partial (y,z)=3$. Fix $x\in X$, and let ${\Gamma }_2^2$ denote the graph with vertex set $\tilde{X}=\{y\in X\mid \partial (x,y)=2\}$ and edge set $\tilde{R}=\{yz\mid y,z\in \tilde{X},\partial (y,z)=2\}$. We show that the adjacency matrix of the local graph ${\Gamma }_2^2$ has at most four distinct eigenvalues. We are motivated by the fact that our assumption above holds if $\Gamma$ is $Q$-polynomial.     

Proper connection and size of graphs

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph $G,$ denoted by $\mathrm{pc}\left(G\right)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Our main result is the following: Let $G$ be a connected graph of order $n$ and $k\ge 2$. If $\left|E\left(G\right)\right|\ge \left(\genfrac{}{}{0ex}{}{n?k?1}{2}\right)+k+2$, then $\mathrm{pc}\left(G\right)\le k$ except when $k=2$ and $G\in \{G_1,G_2\},$ where $G_1=K_1\vee (2K_1+K_2)$ and $G_2=K_1\vee (K_1+2K_2).$     

On disjoint matchings in cubic graphs

For $i=2,3$ and a cubic graph $G$ let ${\nu }_i\left(G\right)$ denote the maximum number of edges that can be covered by $i$ matchings. We show that ${\nu }_2\left(G\right)\ge \frac{4}{5}\left|V\left(G\right)\right|$ and ${\nu }_3\left(G\right)\ge \frac{7}{6}\left|V\left(G\right)\right|$. Moreover, it turns out that ${\nu }_2\left(G\right)\le \frac{\left|V\left(G\right)\right|+2{\nu }_3\left(G\right)}{4}$.     

The Laplacian spectral radius of trees and maximum vertex degree

Let $\Delta \left(T\right)$ and $\mu \left(T\right)$ denote the maximum degree and the Laplacian spectral radius of a tree $T$, respectively. In this paper we prove that for two trees $T_1$ and $T_2$ on $n(n\ge 21)$ vertices, if $\Delta \left(T_1\right)>\Delta \left(T_2\right)$ and $\Delta \left(T_1\right)\ge ?\frac{11n}{30}?+1$, then $\mu \left(T_1\right)>\mu \left(T_2\right)$, and the bound “$\Delta \left(T_1\right)\ge ?\frac{11n}{30}?+1$” is the best possible. We also prove that for two trees $T_1$ and $T_2$ on $2k(k\ge 4)$ vertices with perfect matchings, if $\Delta \left(T_1\right)>\Delta \left(T_2\right)$ and $\Delta \left(T_1\right)\ge ?\frac{k}{2}?+2$, then $\mu \left(T_1\right)>\mu \left(T_2\right)$.     

Un contre-exemple pour un espace d'interpolation qui n'est pas faiblement LUR

According to a previous result of the author, if $(A_0,A_1)$ is an interpolation couple, if $A_0^{?}$ is weakly LUR, then the complex interpolation spaces ${(A_0^{?},A_1^{?})}_{\theta }$ have the same property.Here we construct an interpolation couple $(B_0,B_1)$ where $B_0$ is LUR, but where the complex interpolation spaces ${(B_0,B_1)}_{\theta }$ are not strictly convex.