Stabilization of solutions of a degenerate nonlinear diffusion problem

Abstract

(I) [ 4 = (u% + f(u) in (-L,L) X R+, u(tL,t)=O in R+, u(x, 0) = uo(x) in [-L,L], where m > 1 is a parameter, f is locally Lipschitz continuous, f(0) = 0, and u. is bounded. Problems of this form arise in a number of areas of science; for instance, in models for gas or fluid flow in porous media [2] and for the spread of certain biological populations [13, 161. This paper is divided into two parts. In part I we consider what may be called the motivating example, problem I*, which consists of problem I with the special choice f(u) = U(1 U)(U a) (1) for suitably restricted parameters a. We begin by describing in detail the set % = %(L) of nonnegative equilibrium solutions of problem I*. Clearly 8(L) contains the trivial solution u = 0 for all L > 0. Write 8*(L) = E(L)\(O). In the description of Z*(L) there are two critical parameter values Lo and L1 with 0 < Lo < L1 < + =J. We show that: (i) g*(L) = 4 for 0 < L < LO;