The uniqueness of indefinite nonlinear diffusion problem in population genetics, part I

We study the following Neumann problem in one dimension. {u(t) = du '' + g(x)u(2) (1 - u) in (0,1) x (0, infinity), 0 <= u <= 1 in (0,1) x (0, infinity), u'(0, t) = u'(1, t) = 0 in (0, infinity), g changes sign in (0, 1). This equation models the "complete dominance" case in population genetics of two alleles. It is known that this equation has a nontrivial steady state u(d) for d sufficiently small. We show that the steady state ud is linearly stable. Moreover, under the condition integral(1)(0) g(x) dx >= 0, we show that ud is a unique nontrivial steady state. A conjecture of Nagylaki and Lou in one dimensional case has been largely resolved. (C) 2016 Elsevier Inc. All rights reserved.