Invariant Sets for Nonlinear Elliptic and Parabolic Systems.
Abstract
Systems of coupled second order parabolic equations, or reaction-diffusion equations, arise in the mathematical models of various physical, chemical and biological processes. They describe the evolution in time of two or more substances which interact and diffuse. More specifically, reaction-diffusion equations arise in chemical reactor theory (where the components are concentrations of chemicals), in ecology (densities of species), in the theory of combustion (densities of fuel and thermal energy), and in the theory of nerve impulse transmission (densities of chemicals and electric charges). At any instant of the the densities, which are functions of the space variables, describe the state of the system. An invariant set S is a collection of states with the property that once the system is in a state which is a member of S then, however else the state will evolve, it will at all later times still be a member of S. Symbolically, if U(t) is the state of the time t then U(t sub 0) an element of S implies U(t) an element of S for all t > t sub 0. In this paper criteria are found which can be used to find invariant sets which are described in terms of upper and lower bounds for the various components of the state. As a by-product existence and comparison results for the associated elliptic problems are obtained. In all results both the equations and the boundary conditions are allowed to be nonlinear. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1979
- Accession Number
- ADA068931
Entities
People
- Hendrik J. Kuiper
Organizations
- University of Wisconsin–Madison