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An indefinite nonlinear diffusion problem in population genetics, I: Existence and limiting profiles
An indefinite nonlinear diffusion problem in population genetics, II: Stability and multiplicity
1.  Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210 
2.  School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 
3.  School of Mathematics, University of Minnesota, Minneapolis, MN 55455 
$ u_{t}=d\Delta u+g(x)f(u),~0\leq u\leq 1 $ in Ω × (0, ∞),
$ \frac{\partial u}{\partial\nu}=0 $ on ∂ Ω × (0, ∞),
where $\Delta$ denotes the Laplace operator, $g$ may change sign in $\Omega$, and $f(0)=f(1)=0$, $f(s)>0$ for $s\in(0,1)$. Our main results include stability/instability of the trivial steady states $u\equiv 0$ and $u\equiv 1$, and the multiplicity of nontrivial steady states. This is a continuation of our work [12]. In particular, the conjecture of Nagylaki and Lou [11, p. 152] has been largely resolved. Similar results are obtained for Dirichlet and Robin boundary value problems as well.
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