Abstract
This paper is devoted to the analysis of the large-time behavior of solutions
of one-dimensional Fisher-KPP reaction-diffusion equations. The initial
conditions are assumed to be globally front-like and to decay at infinity
towards the unstable steady state more slowly than any exponentially decaying
function. We prove that all level sets of the solutions move infinitely fast as
time goes to infinity. The locations of the level sets are expressed in terms
of the decay of the initial condition. Furthermore, the spatial profiles of the
solutions become asymptotically uniformly flat at large time. This paper
contains the first systematic study of the large-time behavior of solutions of
KPP equations with slowly decaying initial conditions. Our results are in sharp
contrast with the well-studied case of exponentially bounded initial
conditions.
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