The uniqueness of an indefinite nonlinear diffusion problem in population genetics. II
The paper is concerned with the study of reaction-difusion equation \[ u_t=du''+g(x) u^2(1-u), 0\leq u\leq 1, \quad\text{ in }(0,1)\times (0,\infty) \] subject to homogeneous Neumann boundary conditions \(u'(0,t)=u'(1,t)=0\) in \((0,\infty)\). It is assumed that \(g\) may have a change in sign or even to have multiple zeros over the interval \((0,1)\). Under further suitable conditions on \(g(x)\) the author obtains the uniqueness of a solution and establishes its asymptotic behaviour. It is derived that on compact subset of the set where \(g\) is positive, the unique solution behaves like \(d\) while on compact subset of the set where \(g\) is negative, the unique solution behaves like \(1-a\exp(-b/\sqrt{d})\).
scientific article; zbMATH DE number 6806987