Existence and comparison theorems for nonlinear diffusion systems

Abstract: This article consists of a survey of results concerning the qualitative behavior of solutions of systems of ordinary differential equations which generate an order preserving flow We restrict our consideration to partial orderings on $R^n $ induced by any one of its orthants; a flow preserves ordering if any two solutions $x(t)$ and $y(t)$ are ordered, $x(t) \leqq y(t)$, for all $t > 0$ whenever $x(0) \leqq y(0)$ Many of the important results for such systems have only recently been obtained, principally by M W Hirsch, who pointed out the tendency of their solutions to converge to equilibrium Less well known are some global geometric constraints on the stable manifold of an equilibrium and the existence of heteroclinic orbits connecting ordered equilibria A particularly striking result for this class of systems is the easily computable necessary and sufficient condition for stability of an equilibriumOne of our main goals is to show that by allowing partial orderings on $R^n $ generated by orthants

TL;DR: In this article, a qualitative analysis for a coupled system of two reaction-diffusion equations under various boundary conditions which arises from a number of physical problems is presented, where the nonlinear reaction functions are classified into three basic types according to their relative quasi-monotone property.

TL;DR: In this paper, conditions are given for the Markov jump process describing the stochastic model of chemical reactions with diffusion converges to the solution of the corresponding deterministic reaction-diffusion equation.

TL;DR: In this paper, the authors consider the problem of finding solutions to elliptic boundary value problems in Spaces of Analytic Functions and of Class Mk Generalizations in the case of distributions and Ultra-Distributions.

TL;DR: The One-Dimensional Maximum Principle (MDP) as mentioned in this paper is a generalization of the one-dimensional maximum principle (OMP) for the construction of hyperbolic equations.