Spectrum and Eigenfunctions of a Differential Operator Arising by Linearization of the Fisher and Related Equations
E. J. M. Veling · IMA Journal of Applied Mathematics · 1983
For a class of semilinear diffusion problems from population genetics the linearized differential equation is studied in order to estimate the rate of exponential convergence to some stable stationary solution. Some monotonicity properties of the lowest eigenvalue with respect to the parameters of the problem are given. Two types of lower bounds for this eigenvalue are constructed and compared. For the Fisher non-linearity it turns out that the eigenvalue problem can be solved by an explicit representation of the eigenfunction as a hypergeometric polynomial. For the cubic non-linearity the eigenfunction can be represented by a Heun function.