Nonuniqueness of an indefinite nonlinear diffusion problem in population genetics

We study the following Neumann problem in one-dimension space arising from population genetics: {u_t=du_xx+h(x)u²(1−u)in(−1,1)×(0,∞),0≤u≤1in(−1,1)×(0,∞),u^′(−1,t)=u^′(1,t)=0in(0,∞), where h changes sign in (−1,1) and d is a positive parameter. Lou and Nagylaki (2002) [6] conjectured that if ∫₋₁¹h(x)dx≥0, then this problem has at most one nontrivial steady state (i.e., a time-independent solution which is not identically equal to zero or one). Nakashima (2018) [15] proved this uniqueness under some additional conditions on h(x). Unexpectedly, in this paper, we discover 3 nontrivial steady states for some h(x) satisfying ∫₋₁¹h(x)dx≥0. Moreover, bi-stable phenomenon occurs in this scenario: one with two layers is stable; two with one layer each are ordered with the smaller one being stable and the larger one being unstable.