Asymptotic Solutions of a Model Diffusion-Reaction Equation
R. E. Grundy · IMA Journal of Applied Mathematics · 1988
In this paper we study the large-time solution of the nonlinear diffusion reaction equation ∂u/∂t=(∂2(um)/∂x2)-up,m>1,p>0 , m>1, p>0 subject to u (0, t )=α and initial data with finite support. If we regard the steady state as the leading term in an asymptotic expansion of the solution as t →∞ then we show that this expansion is non-uniform in x . The nature of the non-uniformity, located at the moving interface, is shown to depend crucially on the location in ( p, m ) parameter space. For p <1< m we construct a uniformly valid solution using strained coordinates. In the remaining regions uniform zeroth-order composite solutions are constructed via matched expansions.