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

We study the following Neumann problem in one dimension.{ut=du″+g(x)u2(1−u)in(0,1)×(0,∞),0≤u≤1in(0,1)×(0,∞),u′(0,t)=u′(1,t)=0in(0,∞), 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 ud for d sufficiently small. We show that the steady state ud is linearly stable. Moreover, under the condition ∫01g(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.