Large Time Behavior of Solutions of Systems of Nonlinear Reaction-Diffusion Equations

Abstract

We discuss the asymptotic behavior of solutions of weakly coupled parabolic equations describing systems undergoing diffusion, convection and nonlinear interaction in a bounded spatial domain, fl. If the system admits a bounded invariant region E of phase space then we isolate a parameter r which depends upon the size of 11, the lower bound for the diffusion matrix, the magnitude of convection and a measure of the strength or sensitivity of the reaction. For r > 0 we show that every solution with initial values in E and subject to homogeneous Neumann boundary conditions decays exponentially to a spatially homogeneous function of time. This limiting function is a solution of an ordinary differential equation whose o)-limit sets are determined by the reaction mechanism alone. This result may be interpreted as giving a sufficient condition for the validity of the "lumped parameter" approximation of distributed systems by solutions of ordinary differential equations. In particular we show that all attractors of the ODE are stable as solutions of the PDE as well.

Asymptotic pseudotrajectories and chain recurrent flows, with applications

“…It is interesting to contrast these results with those derived from Conway et al (1978). These assume that the PDE has a compact invariant region Fc R", meaning that Sty(H) c F provided v(H) c F.…”

“…Hale (1986) and Conway et al (1978), for more general equations, give conditions ensuring that solutions to (15), (16) decay to spatially homogeneous solutions, which are interpreted as trajectories of g. Hale considers solutions having initial values in the basin of an attractor for g, while Conway et al take initial values to be in an invariant region for the PDE. In both cases they conclude that u(x, t) is asymptotic with the solution of a certain asymptotically autonomous equation with limit equation dy/dt = g( y ).…”

Scale-invariance in reaction-diffusion models of spatial pattern formation.

“…For instance, in a one-dimensional system of length L, an = n r, and therefore the intervals are disjoint for all n such that jr Sn < ( , .1 [8] Now consider what happens as L increases, perhaps due to growth or to rearrangement of the cells. If L is sufficiently small and all ZDi are positive, all nonuniform disturbances decay exponentially in time and the system returns to a uniform state (4,11). The smallest length at which cs loses stability is LT and the growing mode at this point is 01, which generally has a simple spatial structure.…”

Familles de convexes invariantes et équations de diffusion-réaction

“…-Rappelons que lorsque le coefficient de diffusion a est suffisamment grand, on sait d'après [2] que la solution U tend vers une solution homogène en x située sur le cercle unité. Le résultat qu'on a obtenu ici est plus faible mais est indépendant du coefficient a ; on voit que la solution ne perd pas son homogénéité et ne s'éloigne pas du cercle unité.…”