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.     

Blow-up analysis in quasilinear reaction–diffusion problems with weighted nonlocal source

In this paper, we consider the blow-up of solutions to a class of quasilinear reaction–diffusion problems

where $\Omega$ is a bounded convex domain in ${\mathbb{R}}^n(n\ge 2)$, weighted nonlocal source satisfies $a\left(x\right)f\left(u(x,t)\right)\le a_1+a_2{\left(u(x,t)\right)}^p{\left({\int }_{\Omega }{\left(u(x,t)\right)}^l\mathrm{d}x\right)}^m,$ and $a_1,a_2,p,l,$ and $m$ are positive constants. By utilizing a differential inequality technique and maximum principles, we establish conditions to guarantee that the solution remains global or blows up in a finite time. Moreover, an upper and a lower bound for blow-up time are derived. Furthermore, two examples are given to illustrate the applications of obtained results.     

Innovation based on Gaussian elimination to compute generalized inverse AT,S(2)

In this paper, we execute elementary row and column operations on the partitioned matrix $\left(\begin{array}{cc}\hfill GAG\hfill & \hfill G\hfill \\ \hfill G\hfill & \hfill 0\hfill \end{array}\right)$ into $\left(\begin{array}{cc}\hfill \left(\begin{array}{cc}\hfill I_s\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill ?A_{T,S}^{\left(2\right)}\hfill \end{array}\right)$to compute generalized inverse $A_{T,S}^{\left(2\right)}$ of a given complex matrix $A$, where $G$ is a matrix such that $R\left(G\right)=T$ and $N\left(G\right)=S$. The total number of multiplications and divisions operations is $T(m,n,s)=2m{n}^2+\frac{4m?s?1}{2}ns+(m?s)ns+mns$ and the upper bound of $T(m,n,s)$ is less than $6m{n}^2?\frac{3}{2}{n}^3?\frac{1}{2}{n}^2$ when $n\le m$. A numerical example is shown to illustrate that this method is correct.     

Global and blow-up solutions for nonlinear parabolic equations with Robin boundary conditions

In this paper we discuss the blow-up for classical solutions to the following class of parabolic equations with Robin boundary condition: $\{\begin{array}{cc}{\left(b\left(u\right)\right)}_t=??\left(g\left(u\right)?u\right)+f\left(u\right)\hfill & \text{in }\Omega \times (0,T)\text{,}\hfill \\ \frac{?u}{?n}+\gamma u=0\hfill & \text{on }?\Omega \times (0,T)\text{,}\hfill \\ u(x,0)=h\left(x\right)\ge 0\hfill & \text{in }\overline{\Omega }\text{,}\hfill \end{array}$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N(N\ge 2)$ with smooth boundary $?\Omega$. By constructing some appropriate auxiliary functions and using a first-order differential inequality technique, we derive conditions on the data which guarantee the blow-up or the global existence of the solution. For the blow-up solution, a lower bound on blow-up time is also obtained. Moreover, some examples are presented to illustrate the applications.     

Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity

This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega ?{\mathbb{R}}^2$, where the parameters $\chi ,\mu >0$ and $r\in \mathbb{R}$. Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.     

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.     

Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev exponent

In this work, we are interested in considering the following nonlocal problem

where $\Omega ?{\mathbb{R}}^N(N\ge 4)$ is a smooth bounded domain, $a\ge 0,b>0,1<q<2,\mu ,\lambda >0$ and $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent. By using the variational method and the critical point theorem, some existence and multiplicity results are obtained.     

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.     

Dichotomy of solutions to discrete p-Laplace equations and p-Laplace parabolic equations

In this paper, we discuss and answer the following dichotomy problems: Let $S$ be a network and ${\Delta }_{p,\omega }$ be a discrete $p$-Laplace operator with $1<p<\infty$.(i) If $u,v$ are functions satisfying

then either $u\equiv v$ on $\overline{S}$ or $u<v$ in $S$.(ii) If $u,v$ are functions satisfying

then either $u\equiv v$ on $\overline{S}$ or $u<v$ in $S\times (0,T)$.We believe that this work is not only interesting in itself, but also gives a clue to solve the problems defined on the continuous domain.     

Existence of positive ground state solutions for a critical Kirchhoff type problem with sign-changing potential

In this work, we are interested in studying the following Kirchhoff type problem

where $\Omega ?{\mathbb{R}}^N(N\ge 3)$ is a smooth bounded domain, $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent, $0<q<1,\lambda >0$, and $f\in L^{\infty }\left(\Omega \right)$ with the set $\{x\in \Omega :f\left(x\right)>0\}$ of positive measures, and $g\in L^{\infty }\left(\Omega \right)$ with $g\left(x\right)\ge 0,g?0.$ By the Nehari method and variational method, the existence of positive ground state solutions is obtained.     

Global bifurcation of positive radial solutions for an elliptic system in reactor dynamics

In this paper, we are concerned with the existence of positive radial solutions of the elliptic system $\{\begin{array}{c}?\Delta u=uv?\lambda u+f(\left|x\right|,u)\text{,}{R}_1<\left|x\right|<R_2\text{,}x\in {\mathbb{R}}^N,N\ge 1\text{,}\hfill \\ ?\Delta v=\mu u\text{,}{R}_1<\left|x\right|<R_2\text{,}x\in {\mathbb{R}}^N,N\ge 1\text{,}\hfill \\ u=v=0\text{,}\text{on }\left|x\right|=R_1\text{ and }\left|x\right|=R_2\text{,}\hfill \end{array}$ where $\left|x\right|={\left({\sum }_{i=1}^N{x}_i^2\right)}^{\frac{1}{2}}$, $\lambda >0$ is a constant, $\mu >0$ is a parameter and $0<R_1<R_2<\infty$, $f:[R_1,R_2]\times [0,\infty )\to [0,\infty )$ is continuous and $f(t,s)>0$ for all $(t,s)\in [R_1,R_2]\times (0,\infty )$. Under some appropriate conditions on the nonlinearity $f$, we show that the above system possesses at least one positive radial solution for any $\mu \in (0,\infty )$. The proof of our main results is based upon bifurcation techniques.     

Energy decay rate for a von Karman system with a boundary nonlinear delay term

In this paper, we show the energy decay rate for a von Karman system with a boundary nonlinear delay term. This work is devoted to investigate the influence of kernel function $g$ and the effect of the boundary nonlinear term ${\mu }_1{\left|u_t\left(t\right)\right|}^{m?1}{u}_t\left(t\right)$, a boundary nonlinear time delay term ${\mu }_2{\left|u_t(t?\tau )\right|}^{m?1}{u}_t(t?\tau )$ and prove energy decay rates of solutions when $g$ do not necessarily decay exponentially and the boundary condition has a time delay.     

Existence of ground state solutions of Nehari–Pohozaev type for fractional Schrödinger–Poisson systems with a general potential

In this paper, we consider the following fractional Schrödinger–Poissonproblem

where $0<s\le t<1$ and $2s+2t>3$, the potential $V\left(x\right)$ is weakly differentiable and $f\in C(\mathbb{R},\mathbb{R})$. By introducing some new tricks, we prove that the problem admits a ground state solution of Nehari–Pohozaev type under mild assumptions on $V$ and $f$. The results here extend the existing study.