Propagation of solutions to the Fisher-KPP equation with slowly decaying initial data

The Fisher-KPP equation is a model for population dynamics that has generated a huge amount of interest since its introduction in 1937. The speed with which a population spreads has been computed quite precisely when the initial data, u₀, decays exponentially. More recently, though, the case when the initial data decays more slowly has been studied. In Hamel F and Roques L (2010 J. Differ. Equ. 249 1726-45), the authors show that the level sets of height of m of u move super-linearly and may be bounded above and below by expressions of the form u₀^-1 (c_me^-1) when u₀ decays algebraically of a small enough order. The constants c_m for the upper and lower bounds that they obtain are not explicit and do not match. In this paper, we improve their precision for a broader class of initial data and for a broader class of equations. In particular, our approach yields the explicit highest order term in the location of the level sets, which in the most basic setting is given by as u₀^-1 (me^-t/(1-m)) long as u₀ decays slower than e^-√x. We generalize this to the previously unstudied setting when the nonlinearity is periodic in space. In addition, for large times, we characterize the profile of the solution in terms of a generalized logistic equation.