Steady-state reaction-diffusion systems

Abstract

The steady state for simple reactions is described mathematically by systems of nonlinear ordinary differential equations of the form ϵu_i″=k_ih(u₁,…,u_n,x), where h is nonnegative for u₁≥0,…, u_n≥0. Here ϵ is proportional to the reciprocal of the reaction rate, so for fast reactions is small. We study the behavior of nonnegative solutions of such systems with Dirichlet boundary data in the limit as ϵ→0+. Assuming that h is continuous (but not necessarily differentiable), increasing in u₁, …, u_n, and has the property that u_i = 0 implies h = 0, we show that each concentration u_i tends from above to a suitable piece-wise-linear limiting solution. In particular, if n = 2 and h(u₁,u₂,x) > 0 for u₁u₂>0, the limiting solution fails to be differentiable at only one point; that is, in the limit the reaction occurs only near a single point. But if only h(u₁,u₂,x)≥0 for u₁u₂0 holds, then the limiting solution may fail to be differentiable at two points.