The uniqueness of indefinite nonlinear diffusion problem in population genetics. I. (Q324108)

This paper is concerned with the study of following 1D Neumann problem: NEWLINE\[NEWLINE du''+g(x)u^2(1-u)=0, \quad\text{ in }(0,1), NEWLINE\]NEWLINE NEWLINE\[NEWLINE u'(0)=u'(1)=0 NEWLINE\]NEWLINE and \(g\) satisfies the non-degeneracy condition: whenever \(g(x_0)=0\) then \(x_0\in (0,1)\) and \(g'(x_0)\neq 0\). For small values of \(d>0\) the author obtains that the above problem admits a linearly stable solution \(u_d\) which is unique if \(\int_0^1 g(x)dx\geq 0\). Further, it is shown that if \(\int_0^1 g(x)dx<0\) then any solution \(v\) of the above problem satisfies either \(v=u_d\) or \(v\leq Cd\) in \([0,1]\) for some \(C>0\).