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 = d u x x + h ( x ) u 2 ( 1 − u ) in ( − 1 , 1 ) × ( 0 , ∞ ) , 0 ≤ u ≤ 1 in ( − 1 , 1 ) × ( 0 , ∞ ) , u ′ ( − 1 , t ) = u ′ ( 1 , t ) = 0 in ( 0 , ∞ ) , where h changes sign in ( − 1 , 1 ) and d is a positive parameter. Lou and Nagylaki (2002) [6] conjectured that if ∫ − 1 1 h ( x ) d x ≥ 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 ∫ − 1 1 h ( x ) d x ≥ 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.