Take the linearised FKPP equation ∂th=∂2xh+h∂th=∂x2h+h with boundary condition h(m(t), t) = 0. Depending on the behaviour of the initial condition h 0(x) = h(x, 0) we obtain the asymptotics—up to a o(1) term r(t)—of the absorbing boundary m(t) such that ω(x):=limt→∞h(x+m(t),t)ω(x):=limt→∞h(x+m(t),t) exists and is non-trivial. In particular, as in Bramson’s results for the non-linear FKPP equation, we recover the celebrated −3/2logt−3/2logt correction for initial conditions decaying faster than xνe−xxνe−x for some ν<−2ν<−2. Furthermore, when we are in this regime, the main result of the present work is the identification (to first order) of the r(t) term, which ensures the fastest convergence to ω(x)ω(x). When h 0(x) decays faster than xνe−xxνe−x for some ν<−3ν<−3, we show that r(t) must be chosen to be −3π/t−−−√−3π/t, which is precisely the term predicted heuristically by Ebert–van Saarloos (Phys. D Nonlin. Phenom. 146(1): 1–99, 2000) in the non-linear case (see also Mueller and Munier Phys Rev E 90(4):042143, 2014, Henderson, Commun Math Sci 14(4):973–985, 2016, Brunet and Derrida Stat Phys 1-20, 2015). When the initial condition decays as xνe−xxνe−x for some ν∈[−3,−2)ν∈[−3,−2), we show that even though we are still in the regime where Bramson’s correction is −3/2logt−3/2logt, the Ebert–van Saarloos correction has to be modified. Similar results were recently obtained by Henderson CommunMath Sci 14(4):973–985, 2016 using an analytical approach and only for compactly supported initial conditions.
Berestycki, J., Brunet, É., Harris, S., & Roberts, M. (2017). Vanishing corrections for the position in a linear model of FKPP fronts. Communications in Mathematical Physics, 349(3), 857-893. https://doi.org/10.1007/s00220-016-2790-9