Finite time blow-up for a class of parabolic or pseudo-parabolic equations

In this paper, we study the initial boundary value problem for a class of parabolic or pseudo-parabolic equations:

$u_t?a\Delta u_t?\Delta u+bu=k\left(t\right)|u{|}^{p?2}u,(x,t)\in \Omega \times (0,T),$

where $a\ge 0$, $b>?{?}_1$ with ${?}_1$ being the principal eigenvalue for $?\Delta$ on $H_0^1\left(\Omega \right)$ and $k\left(t\right)>0$. By using the potential well method, Levine’s concavity method and some differential inequality techniques, we obtain the finite time blow-up results provided that the initial energy satisfies three conditions: (i) $J(u_0;0)<0$; (ii) $J(u_0;0)\le d\left(\infty \right)$, where $d\left(\infty \right)$ is a nonnegative constant; (iii) $0<J(u_0;0)\le C\rho \left(0\right)$, where $\rho \left(0\right)$ involves the $L^2$-norm or $H_0^1$-norm of the initial data. We also establish the lower and upper bounds for the blow-up time. In particular, we obtain the existence of certain solutions blowing up in finite time with initial data at the Nehari manifold or at arbitrary energy level.     

The Laplacian spectral radius of bicyclic graphs with a given girth

Let $?(n,g)$ be the class of bicyclic graphs on $n$ vertices with girth $g$. Let ${?}_1(n,g)$ be the subclass of $?(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and ${?}_2(n,g)=?(n,g)?{?}_1(n,g)$. This paper determines the unique graph with the maximal Laplacian spectral radius among all graphs in ${?}_1(n,g)$ and ${?}_2(n,g)$, respectively. Furthermore, the upper bound of the Laplacian spectral radius and the extremal graph for $?(n,g)$ are also obtained.     

Trees with minimal Laplacian coefficients

Let $G$ be a simple undirected graph with the characteristic polynomial of its Laplacian matrix $L\left(G\right)$, $P(G,\mu )={\sum }_{k=0}^n{(?1)}^k{c}_k{\mu }^{n?k}$. It is well known that for trees the Laplacian coefficient $c_{n?2}$ is equal to the Wiener index of $G$, while $c_{n?3}$ is equal to the modified hyper-Wiener index of the graph. In this paper, we characterize $n$-vertex trees with given matching number $m$ which simultaneously minimize all Laplacian coefficients. The extremal tree $A(n,m)$ is a spur, obtained from the star graph $S_{n?m+1}$ with $n?m+1$ vertices by attaching a pendant edge to each of certain $m?1$ non-central vertices of $S_{n?m+1}$. In particular, $A(n,m)$ minimizes the Wiener index, the modified hyper-Wiener index and the recently introduced Incidence energy of trees, defined as $IE\left(G\right)={\sum }_{k=0}^n\sqrt{{\mu }_k}$, where ${\mu }_k$ are the eigenvalues of signless Laplacian matrix $Q\left(G\right)=D\left(G\right)+A\left(G\right)$. We introduced a general $\rho$ transformation which decreases all Laplacian coefficients simultaneously. In conclusion, we illustrate on examples of Wiener index and Incidence energy that the opposite problem of simultaneously maximizing all Laplacian coefficients has no solution.     

A new approach toward stabilization in a two-species chemotaxis model with logistic source

This paper aims at providing an alternative approach to study global dynamic properties for a two-species chemotaxis model, with the main novelty being that both populations mutually compete with the other on account of the Lotka–Volterra dynamics. More precisely, we consider the following Neumann initial–boundary value problem

in a bounded domain $\Omega ?{\mathbb{R}}^n,n\ge 1$, with smooth boundary, where $d_1,d_2,d_3,{\chi }_1,{\chi }_2,{\mu }_1,{\mu }_2,a_1,a_2$ are positive constants.When $a_1\in (0,1)$ and $a_2\in (0,1)$, it is shown that under some explicit largeness assumptions on the logistic growth coefficients ${\mu }_1$ and ${\mu }_2$, the corresponding Neumann initial–boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state $(u^1,v^1,w^1)$ of above system in the large time limit. The respective decay rate of this convergence is shown to be exponential.When $a_1\ge 1$ and $a_2\in (0,1)$, if ${\mu }_2$ is suitable large, for all sufficiently regular nonnegative initial data $u_0$ and $v_0$ with $u_0?0$ and $v_0?0$, the globally bounded solution of above system will stabilize toward $(0,1,1)$ as $t\to \infty$ in algebraic.     

Longest paths and cycles in faulty star graphs

In this paper, we investigate the star graph $S_n$ with faulty vertices and/or edges from the graph theoretic point of view. We show that between every pair of vertices with different colors in a bicoloring of $S_n$, $n⩾4$, there is a fault-free path of length at least $n!-2f_v-1$, and there is a path of length at least $n!-2f_v-2$ joining a pair of vertices with the same color, when the number of faulty elements is $n-3$ or less. Here, $f_v$ is the number of faulty vertices. $S_n$, $n⩾4$, with at most $n-2$ faulty elements has a fault-free cycle of length at least $n!-2f_v$ unless the number of faulty elements are $n-2$ and all the faulty elements are edges incident to a common vertex. It is also shown that $S_n$, $n⩾4$, is strongly hamiltonian-laceable if the number of faulty elements is $n-3$ or less and the number of faulty vertices is one or less.     

Convergence for a family of discrete advection–reaction operators

We define a family of discrete Advection–reaction operators, denoted by $A_{a\lambda }$, which associate to a given scalar sequence $s=\left\{s_n\right\}$ the sequence given by $A_{a\lambda }\left(s\right)\equiv \left\{b_n\right\}$, where $b_n=a_{n?2}{s}_{n?1}+{\lambda }_n{s}_n$ for $n=1,2,\dots$. For $A_{a\lambda }$ we explicitly find their iterates and study their convergence properties. Finally, we show the relationship between the family of discrete operators with the continuous one dimensional advection–reaction equation.