A uniform solution of a parabolic problem with angular layer behavior

Abstract

For a singularly perturbed linear parabolic partial differential equation with a C1-discontinuous initial data, the solution possesses an angular layer behavior. We construct an angular layer equation subject to auxiliary conditions. The angular layer function can be written in terms of the first integral of the complementary error function after a careful analysis on the C1-jump discontinuity of the outer solution. An asymptotic expansion obtained is shown to be uniformly valid with the first order accuracy in the small parameter by employing the maximum principle and some exponentially decay estimate of the angular layer function.

Keywords:

• singular perturbation
• internal layer
• angular layer
• method of matched asymptotic expansions
• maximum principle