Steady-state reaction-diffusion-convection equations: dead cores and singular perturbations

Abstract

(here E > 0, f(0) = 0, and f(u) > 0 for II > 0) or generalizations have been considered by several authors (Bandle, Sperb, and Stakgold [l], Bobisud [2], Bobisud and Stakgold [3], Chang and Howes [I], Fife [5], Friedman and Phillips [6], O’Malley [S, 91, Stakgold [lo], among others). Let z(x) be the solution of the degenerate equation obtained by formally setting E = 0 in (l)-in this case, z(x) = 0. Then Bobisud [2], Chang and Howes [J]. Fife [5], and O’Malley [S, 91 are concerned with the singular perturbation problem, that is, with showing that u,(x) ---, z(x) as E + 0, uniformly for x in a compact subset of (1, l), and with estimating the order of this degeneration. On the other hand, Bandle, Sperb, and Stakgold [I]. Bobisud [2], Bobisud and Stakgold [3], Friedman and Phillips [6], and Stakgold [lo] show a stronger result when f is not Lipschitz continuous at 0: given any compact subset A C (1, l), u,(x) = z(x) on A for all sufficiently small E. For any E > 0, the subset of [1. I] on which u,(x) = z(x) is called the dead core for that E, a terminology that stems from applications to chemical reactors. The special feature of z(x) that makes the existence of a dead core possible is that z is a solution of the full equation EZ” = f(z) for all E 2 0, as will be seen below. Our concern here is the extension of results of both sorts to right hand sides that depend on u; as well as cl,, i.e. to the problem