一类非牛顿渗流系统爆破界的估计

首先得到一类拟线性椭圆型方程组的正解的先验界估计和衰减性质,从而推出该方程组的径向非增正对称解的不存在性结果.利用此结果建立了一类拟线性反应扩散方程组(非牛顿渗流系统)的爆破界的估计,推广了半线性(Fujita型)反应扩散方程组的结果.     

Non-existence of positive solution to a quasilinear elliptic system and blow-up estimates for a reaction-diffusion system

IntroductionIn these years, the reaction-diffusion systems of Fujita typea5 ttell as the related elliptic systemt'ith fl g RN, Tnl. n1 3 0, i = 1, 2. \vere studied by' a nu111ber of authors. The probiemsconcerning system (l) inc1ude globa1 existence and g1obal existen(.e numbers. b1ow-up. bloxv-uprates, and blow-up sets. uniqueness or nonuniqueness. et('. FOr s}'stem (2) there are problemssuch as existence or non-existence. uniqueness or nonllniqueI1ess. and so ol1. 1Ieanwhile. itseems that…     

Blow-up estimates of positive radial solutions of a class first-order evolution systems

We derive the nonexistence of radially positive solutions for a system of quasilinear elliptic differential equations, and establish blow-up estimates for a class of first-order evolution systems (in time). The main results are new and extend previous known results.     

Non-existence of positive radial solutions for a class of quasilinear elliptic system

In this paper, our main purpose is to establish some nonexistence results of positive radial solutions to the quasilinear ordinary differential equation system. The main results of the present paper are new and extend the previously known results.     

Long term dynamics of a reaction-diffusion system

We study a reaction-diffusion system of two parabolic differential equations describing the behavior of a nuclear reactor. We provide existence results for nontrivial periodic solutions, nonexistence results for stationary solutions and we prove that, depending on the value of the parameters, either the system admits a compact global attractor, or the solutions are unbounded.     

Asymptotic behavior of solutions of a periodic diffusion system

This paper deals with a periodic reaction-diffusion system of plankton allelopathy under homogeneous Neumann boundary conditions. Based on the result of Ahmad and Lazer, we show some estimates and nonexistence results for the positive solutions of the system. Furthermore, we investigate the asymptotic behavior of the solutions of the system, that is one species dies out and the other exists as time t tends to infinity.     

Nonlinear oscillations of fourth order quasilinear ordinary differential equations

We consider fourth order quasilinear ordinary differential equations. Firstly, we classify positive solutions into four types according to their asymptotic properties. Then we derive existence theorems of positive solutions belonging to each type. Using these results, we can obtain an oscillation criterion, which is our main objective. Moreover, applying such criteria for ordinary differential equations to binary elliptic systems, we establish nonexistence theorems for positive solutions.     

Non-linear reaction-diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions

This work first considers the classical Lie symmetry analysis of a class of systems of two quasilinear reaction-diffusion equations having variable diffusivities. Subsequently, non-Lie reductions to systems of first order ordinary differential equations are obtained for a subclass of these systems. In particular, families of exact solutions of a diffusive Lotka-Volterra type system are constructed.     

一类非牛顿渗流方程爆破界的估计

本文利用拟线性常微分方程解的非存在性定理得到了一类拟线性反应扩散方程(非牛顿渗流方程)爆破界的估计,从而推广了半线性反应扩散方程(牛顿渗流方程)相应结果.     

Existence and nonexistence results for reaction-diffusion equations in product of cones

Problems of existence and nonexistence of global nontrivial solutions to quasilinear evolution differential inequalities in a product of cones are investigated. The proofs of the nonexistence results are based on the test-function method developed, for the case of the whole space, by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established using the method of supersolutions.     

Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities

This paper deals with a quasilinear parabolic system coupled via both nonlinear reaction terms and nonlinear boundary flux. As the results of the interaction among the multi-coupled nonlinearities in the system, some appropriate conditions for global existence and global nonexistence of solutions are determined respectively.     

Global nonexistence of positive initial energy solution for coupled quasilinear system

In this paper, we consider global nonexistence of a solution for coupled quasilinear system with damping and source under Dirichlet boundary condition. We obtain a global nonexistence result of solution by using the perturbed energy method, where the initial energy is assumed to be positive. Copyright © 2017 John Wiley & Sons, Ltd.     

On a system of second-order multi-point boundary value problems

We study the existence of positive solutions for a system of nonlinear second-order ordinary differential equations subject to some multi-point boundary conditions. The nonexistence of positive solutions is also investigated.     

On positive solution for a class of degenerate quasilinear elliptic positone/semipositone systems

This paper deals with the existence and nonexistence of positive weak solutions of degenerate quasilinear elliptic systems with subcritical and critical exponents. The nonlinearities involved have semipositone and positone structures and the existence results are obtained by applying the lower and upper-solution method and variational techniques.     

On differential inequalities with point singularities on the boundary

We prove the nonexistence of solutions for some nonlinear ordinary differential equations and inequalities, for quasilinear partial differential equations and inequalities in bounded domains with singular points on the boundary, and for systems of such equations and inequalities. The proofs are based on the method of nonlinear capacity. We also give examples demonstrating that the conditions obtained are sharp in the class of problems under consideration.     

Some Notes on Reaction Diffusion Systems with Nonlinear Boundary Conditions

This paper deals with the existence and nonexistence of global positive solution to a semilinear reaction-diffusion system with nonlinear boundary conditions.For the heat diffusion case,the necessary and sufficient conditions on the global existence of all positive solutions are obtained.For the general fast diffusion case,we get some conditions on the global existence and nonexistence of positive solutions.The results of this paper fill the some gaps which were left in this field.     

Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays

This paper is concerned with three 3-species time-delayed Lotka-Volterra reaction-diffusion systems and their corresponding ordinary differential systems without diffusion. The time delays may be discrete or continuous, and the boundary conditions for the reaction-diffusion systems are of Neumann type. The goal of the paper is to obtain some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady-state solution for each of the three model problems. These conditions involve only the reaction rate constants and are independent of the diffusion effect and time delays. The result of global asymptotic stability implies that each of the three model systems coexists, is permanent, and the trivial and all semitrivial solutions are unstable. Our approach to the problem is based on the method of upper and lower solutions for a more general reaction-diffusion system which gives a common framework for the 3-species model problems. Some global stability results for the 2-species competition and prey-predator reaction-diffusion systems are included in the discussion.     

On higher order elliptic and parabolic inequalities with singularities on the boundary

We prove the nonexistence of solutions to a number of higher order quasilinear elliptic and parabolic partial differential inequalities in bounded domains with point singularities on the boundary. The results are extended to systems of such inequalities. The proofs are based on the method of nonlinear capacity. We also present examples showing that the conditions obtained are sharp in the class of problems under consideration.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.