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.     

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.     

State complexity of power

The number of states in a deterministic finite automaton (DFA) recognizing the language $L^k$, where $L$ is regular language recognized by an $n$-state DFA, and $k?2$ is a constant, is shown to be at most $n{2}^{(k?1)n}$ and at least $(n?k)2^{(k?1)(n?k)}$ in the worst case, for every $n>k$ and for every alphabet of at least six letters. Thus, the state complexity of $L^k$ is $\Theta \left(n{2}^{(k?1)n}\right)$. In the case $k=3$ the corresponding state complexity function for $L^3$ is determined as $\frac{6n?3}{8}{4}^n?(n?1)2^n?n$ with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of $L^k$ is demonstrated to be $nk$. This bound is shown to be tight over a two-letter alphabet.     

Relations between matrix sets generated from linear matrix expressions and their applications

Let $A+BXC$ and $A+BX+YC$ be two linear matrix expressions, and denote by $\{A+BXC\}$ and $\{A+BX+YC\}$ the collections of the two matrix expressions when $X$ and $Y$ run over the corresponding matrix spaces. In this paper, we study relationships between the two matrix sets $\{A_1+B_1{X}_1{C}_1\}$ and $\{A_2+B_2{X}_2{C}_2\}$, as well as the two sets $\{A_1+B_1{X}_1+Y_1{C}_1\}$ and $\{A_2+B_2{X}_2+Y_2{C}_2\}$, by using some rank formulas for matrices. In particular, we give necessary and sufficient conditions for the two matrix set inclusions $\{A_1+B_1{X}_1{C}_1\}?\{A_2+B_2{X}_2{C}_2\}$ and $\{A_1+B_1{X}_1+Y_1{C}_1\}?\{A_2+B_2{X}_2+Y_2{C}_2\}$ to hold. We also use the results obtained to characterize relations of solutions of some linear matrix equations.     

Nonexistence of global solutions for a class of nonlocal in time and space nonlinear evolution equations

In this paper, we study the nonlocal nonlinear evolution equation

${}^C{D}_{0|t}^{\alpha }u(t,x)?(J1\left|u\right|?\left|u\right|)(t,x){+}^C{D}_{0|t}^{\beta }u(t,x)={\left|u(t,x)\right|}^p,t>0,x\in {\mathbb{R}}^d,$

where $1<\alpha <2$, $0<\beta <1$, $p>1$, $J:{\mathbb{R}}^d\to {\mathbb{R}}_{+}$, $1$ is the convolution product in ${\mathbb{R}}^d$, and ${}^C{D}_{0|t}^q$, $q\in \{\alpha ,\beta \}$, is the Caputo left-sided fractional derivative of order $q$ with respect to the time $t$. We prove that the problem admits no global weak solution other than the trivial one with suitable initial data when $1<p<1+\frac{2\beta }{d\beta +2(1?\beta )}$. Next, we deal with the system

where $1<\alpha <2$, $0<\beta <1$, $p>1$, and $q>1$. We prove that the system admitsnon global weak solution other than the trivial one with suitable initial data when $1<pq<1+\frac{2\beta }{d\beta +2(1?\beta )}max\{p+1,q+1\}$. Our approach is based on the test function method.     

Existence of local strong solutions for motions of electrorheological fluids in three dimensions

We prove for three-dimensional domains the existence of local strong solutions to systems of nonlinear partial differential equations with $p(\cdot )$-structure, $p_{\infty }\le p(\cdot )\le p_0$, and Dirichlet boundary conditions for $p_{\infty }>\frac{9}{5}$ without restriction on the upper bound $p_0$. In particular this result is applicable to the motion of electrorheological fluids.     

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.     

Suslin’s algorithms for reduction of unimodular rows

A well-known lemma of Suslin says that for a commutative ring $A$ if $(v_1\left(X\right),\dots ,v_n\left(X\right))\in {\left(A\left[X\right]\right)}^n$ is unimodular where $v_1$ is monic and $n\ge 3$, then there exist ${\gamma }_1,\dots ,{\gamma }_{\ell }\in E_{n-1}\left(A\left[X\right]\right)$ such that the ideal generated by $\mathrm{Res}(v_1,e_1.{\gamma }_1{}^t(v_2,\dots ,v_n)),\dots ,\mathrm{Res}(v_1,e_1.{\gamma }_{\ell }{}^t(v_2,\dots ,v_n))$ equals $A$. This lemma played a central role in the resolution of Serre’s Conjecture. In the case where $A$ contains a set $E$ of cardinality greater than $deg{v}_1+1$ such that $y-y^{\prime }$ is invertible for each $y\ne y^{\prime }$ in $E$, we prove that the ${\gamma }_i$ can simply correspond to the elementary operations $L_1\to L_1+y_i{\sum }_{j=2}^{n-1}{u}_{j+1}{L}_j$, $1\le i\le \ell =deg{v}_1+1$, where $u_1{v}_1+\cdots +u_n{v}_n=1$. These efficient elementary operations enable us to give new and simple algorithms for reducing unimodular rows with entries in $K[X_1,\dots ,X_k]$ to ${}^t(1,0,\dots ,0)$ using elementary operations in the case where $K$ is an infinite field. Another feature of this paper is that it shows that the concrete local–global principles can produce competitive complexity bounds.     

Existence of ground states for a Kirchhoff type problem without 4-superlinear condition

We consider the existence of ground state solutions for the Kirchhoff type problem

where $a,b>0$, $N=1,2,3$ and $2<p<2^1$. Here we are interested in the case that $2<p\le 4$ since the existence of ground state for $4<p\le 2^1$ is easily obtained by a standard variational argument. Our method is based on a Poho$\stackrel{?}{z}$aev type identity.