Positive solutions to a class of second order differential equations with singular nonlinearities

A class of second order quasilinear differential equations with singular nonlinearities is considered. The set of all possible solutions defined on a positive half-line [a,∞) is classified into six types according to their aymptotic behavior as t→∞, and sharp conditions are established for the existence of solutions belonging to each of the classified types.     

Positive decreasing solutions of quasilinear dynamic equations

We consider a quasilinear dynamic equation reducing to a half-linear equation, an Emden–Fowler equation or a Sturm–Liouville equation under some conditions. Any nontrivial solution of the quasilinear dynamic equation is eventually monotone. In other words, it can be either positive decreasing (negative increasing) or positive increasing (negative decreasing). In particular, we investigate the asymptotic behavior of all positive decreasing solutions which are classified according to certain integral conditions. The approach is based on the Tychonov fixed point theorem.     

On nonlinear reaction-diffusion systems

This paper presents a qualitative analysis for a coupled system of two reaction-diffusion equations under various boundary conditions which arises from a number of physical problems. The nonlinear reaction functions are classified into three basic types according to their relative quasi-monotone property. For each type of reaction functions, an existence-comparison theorem, in terms of upper and lower solutions, is established for the time-dependent system as well as some boundary value problems. Three concrete physical systems arising from epidemics, biochemistry and engineering are taken as representatives of the basic types of reacting problems. Through suitable construction of upper and lower solutions, various qualitative properties of the solution for each system are obtained. These include the existence and bounds of time-dependent solutions, asymptotic behavior of the solution, stability and instability of nontrivial steady-state solutions, estimates of stability regions, and finally the blowing-up property of the solution. Special attention is given to the homogeneous Neumann boundary condition.     

On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains

The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen–Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.     

Asymptotic forms of positive solutions of third-order Emden-Fowler equations

We obtain asymptotic forms of positive solutions of third-order Emden-Fowler equations. These results improve earlier results for asymptotic behavior of positive solutions considerably.     

Unbounded positive solutions of higher-order nonlinear functional differential equations

Classification schemes for positive solutions of a class of higher-order nonlinear functional differential equations are given in terms of their asymptotic behavior, and necessary as well as sufficient conditions for the existence of these solutions are also provided.     

奇摄动拟线性系统的边界层和角层性质

本文利用微分不等式的方法研究二阶拟线性系统狄立克雷问题解的存在和当ε→0+时它们的渐近性质.根据退化解在(a,b)中是否有连续的一阶偏导数,研究了解的两种渐近形式,从而导出边界层和角层现象.     

Global asymptotic behavior in a Lotka–Volterra competition system with spatio-temporal delays

This paper is concerned with a Lotka–Volterra competition system with spatio-temporal delays. By using the linearization method, we show the local asymptotic behavior of the nonnegative steady-state solutions. Especially, the global asymptotic stability of the positive steady-state solution is investigated by the method of upper and lower solutions. The result of global asymptotic stability implies that the system has no nonconstant positive steady-state solution.     

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.     

On the asymptotic spectra of the bounded solutions of a nonlinear Volterra equation

We consider the asymptotic behavior of the bounded solutions of a nonlinear Volterra integrodifferential equation with a positive definite convolution kernel. Our main result states that (under appropriate assumptions) the asymptotic spectra of the solutions are contained in the set where the real part of the Fourier transform of the kernel vanishes. We also give a new asymptotic stability theorem, and present a new proof of a known result on the asymptotic behavior of the bounded solutions of a nonlinear, nondifferentiated Volterra equation.     

Global attractor of a coupled finite difference reaction diffusion system with delays

In the study of asymptotic behavior of solutions for reaction diffusion systems, an important concern is to determine whether and when the system has a global attractor which attracts all positive time-dependent solutions. The aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays. Sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady-state system. Applications are given to three types of Lotka-Volterra reaction diffusion models, where time-delays may appear in the opposing species.     

ON MULTIPLE POSITIVE SOLUTIONS FOR A NONLINEAR ELLIPTIC PROBLEM IN R^N

61. IntroductionConsider the semilinear elliptic problemwhere p E (1, ac) for N 2 3, and p E (l, co) for N = 1, 2, 0 S a(x) E C(R"), A > 0 is areal parameter.The existence and uniqueness of solution for such problems have been considered by mailyauthors recently (see [l--4] and references therein). In I3], T. Bartsch and Wang, Z. Q. provedthat with more genaral nolilineaxities (1.1) has at least one solution for A large under someconditions on a(x), one of which is that a--1 (0) has non…     

Asymptotic Behavior of Solutions of Dynamic Equations

We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.     

Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone

This paper is concerned with the positive stationary problem of a Lotka–Volterra cross-diffusive competition model with a protection zone for the weak competitor. The detailed structure of positive stationary solutions for small birth rates and large cross-diffusion is shown. The structure is quite different from that without cross-diffusion, from which we can see that large cross-diffusion has a beneficial effect for the existence of positive stationary solutions. The effect of the spatial heterogeneity caused by protection zones is also examined and is shown to change the shape of the bifurcation curve. Thus the environmental heterogeneity, together with large cross-diffusion, can produce much more complicated stationary patterns. Finally, the asymptotic behavior of positive stationary solutions for any birth rate as the cross-diffusion coefficient tends to infinity is given, and moreover, the structure of positive solutions of the limiting system is analyzed. The result of asymptotic behavior also reveals different phenomena from that of the homogeneous case without protection zones.     

Positive solutions to generalized Dickman equation

The paper investigates large-time behaviour of positive solutions to a generalized Dickman equation. The asymptotic behaviour of dominant and subdominant positive solutions is analysed and a structure formula describing behaviour of all solutions is proved. A criterion is also given for sufficient conditions on initial functions to generate positive solutions with prescribed asymptotic behaviour with values of their weighted limits computed.     

Existence and Asymptotic Behavior of Solutions for Quasilinear Parabolic Systems

This paper is concerned with the existence, uniqueness and asymptotic behavior of solutions for the quasilinear parabolic systems with mixed quasimonotone reaction functions endowed with Dirichlet boundary condition, in which the elliptic operators are allowed to be degenerate. By the method of the coupled upper and lower solutions and its monotone iterations, it is shown that a pair of coupled upper and lower solutions ensures that the unique positive solution exists and is globally stable if the quasisolutions are equal. Moreover, we study the asymptotic behavior of solutions to the Lotka-Volterra predator-prey model with the density-dependent diffusion.     

Positive solutions for a diffusive Bazykin model in spatially heterogeneous environment

In this paper, we investigate a diffusive Bazykin model in a spatially heterogeneous environment. We obtain some results on nonexistence and existence of positive solutions of the model. Moreover, the asymptotic behavior of positive solutions with respect to certain parameters is also studied.     

Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses

In this paper, we investigate the dynamics of a class of the so-called semi-ratio-dependent predator-prey interaction models with functional responses based on systems of nonautonomous differential equations with time-dependent parameters. The functional responses are classified into five types and typical examples of each type are provided. Then we establish sufficient criteria for the boundedness of solutions, the permanence of system, and the existence, uniqueness and globally asymptotic stability of positive periodic solution and positive almost periodic solution. Some conclusive discussion is presented at the end of this paper.     

Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria

We consider the asymptotic behavior for large time of solutions to reaction-diffusion systems modeling reversible chemical reactions. We focus on the case where multiple equilibria exist. In this case, due to the existence of so-called "boundary equilibria", the analysis of the asymptotic behavior is not obvious. The solution is understood in a weak sense as a limit of adequate approximate solutions. We prove that this solution converges in L^1 toward an equilibrium as time goes to infinity and that the convergence is exponential if the limit is strictly positive.     

Non-monotone traveling waves and entire solutions for a delayed nonlocal dispersal equation

This paper is concerned with the traveling waves and entire solutions for a delayed nonlocal dispersal equation with convolution- type crossing-monostable nonlinearity. We first establish the existence of non-monotone traveling waves. By Ikehara’s Tauberian theorem, we further prove the asymptotic behavior of traveling waves, including monotone and non-monotone ones. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. Finally, the entire solutions are considered. By introducing two auxiliary monostable equations and establishing some comparison arguments for the three equations, some new types of entire solutions are constructed via the traveling wavefronts and spatially independent solutions of the auxiliary equations.