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.     

Singular discrete and continuous mixed boundary value problems

For each $n\in \mathbb{N}$, $n\ge 2$, we prove the existence of a solution $(u_0,\dots ,u_n)\in {\mathbb{R}}^{n+1}$ of the singular discrete problem $\frac{1}{h^2}{\Delta }^2{u}_{k?1}+f(t_k,u_k)=0\text{,}k=1,\dots ,n?1\text{,}$$\Delta u_0=0\text{,}{u}_n=0\text{,}$ where $u_k>0$ for $k=0,\dots ,n?1$. Here $T\in (0,\infty )$, $h=\frac{T}{n}$, $t_k=hk$, $f(t,x):[0,T]\times (0,\infty )\to \mathbb{R}$ is continuous and has a singularity at $x=0$. We prove that for $n\to \infty$ the sequence of solutions of the above discrete problems converges to a solution $y$ of the corresponding continuous boundary value problem $y^{″}\left(t\right)+f(t,y\left(t\right))=0\text{,}$$y^{\prime }\left(0\right)=0\text{,}y\left(T\right)=0\text{.}$     

On a nonlinear wave equation involving the term −∂∂x(μ(x,t,u,6ux62)ux): Linear approximation and asymptotic expansion of solution in many small parameters

In this paper, we consider the following nonlinear Kirchhoff wave equation (1)$\{\begin{array}{c}{u}_{tt}?\frac{?}{?x}\left(\mu (x,t,u,{6u_{x}6}^2)u_x\right)=f(x,t,u,u_x,u_t)\text{,}0<x<1,0<t<T\text{,}\hfill \\ u(0,t)=g_0\left(t\right)\text{,}u(1,t)=g_1\left(t\right)\text{,}\hfill \\ u(x,0)={\stackrel{?}{u}}_0\left(x\right)\text{,}{u}_t(x,0)={\stackrel{?}{u}}_1\left(x\right)\text{,}\hfill \end{array}$ where ${\stackrel{?}{u}}_0$, ${\stackrel{?}{u}}_1$, $\mu$, $f$, $g_0$, $g_1$ are given functions and ${6u_{x}6}^2={\int }_0^1{u}_x^2(x,t)dx$. First, combining the linearization method for nonlinear term, the Faedo–Galerkin method and the weak compact method, a unique weak solution of problem (1) is obtained. Next, by using Taylor’s expansion of the function $\mu (x,t,y,z)$ around the point $(x,t,y_0,z_0)$ up to order $N+1$, we establish an asymptotic expansion of high order in many small parameters of solution.     

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 $\mu \left(T_1\right)>\mu \left(T_2\right)$.     

The minimum volume of subspace trades

A subspace bitrade of type $T_q(t,k,v)$ is a pair $(T_0,T_1)$ of two disjoint nonempty collections of $k$-dimensional subspaces of a $v$-dimensional space $V$ over the finite field of order $q$ such that every $t$-dimensional subspace of $V$ is covered by the same number of subspaces from $T_0$ and $T_1$. In a previous paper, the minimum cardinality of a subspace $T_q(t,t+1,v)$ bitrade was established. We generalize that result by showing that for admissible $v$, $t$, and $k$, the minimum cardinality of a subspace $T_q(t,k,v)$ bitrade does not depend on $k$. An example of a minimum bitrade is represented using generator matrices in the reduced echelon form. For $t=1$, the uniqueness of a minimum bitrade is proved.     

An explicit counterexample for the Lp-maximal regularity problem

In this short Note we give a self-contained example of a consistent family of holomorphic semigroups $(T_p{(t))}_{t?0}$ such that $(T_p{(t))}_{t?0}$ does not have maximal regularity for $p>2$. This answers negatively the open question whether maximal regularity extrapolates from $L^2$ to the $L^p$-scale.     

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 $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

Positive solution of fourth order ordinary differential equation with four-point boundary conditions

In this work, the authors consider the fourth order nonlinear ordinary differential equation$u^{\left(4\right)}\left(t\right)=f(t,u\left(t\right))\text{,}0<t<1\text{,}$ with the four-point boundary conditions $u\left(0\right)=u\left(1\right)=0\text{,}$$a{u}^{″}\left({\xi }_1\right)-b{u}^{‴}\left({\xi }_1\right)=0\text{,}c{u}^{″}\left({\xi }_2\right)+d{u}^{‴}\left({\xi }_2\right)=0\text{,}$ where $0\le {\xi }_1<{\xi }_2\le 1$. By means of the upper and lower solution method and fixed point theorems, some results on the existence of positive solutions to the above four-point boundary value problem are obtained.