Abstract
Take the linearised FKPP equation \({\partial_{t}h = \partial^{2}_{x}h + h}\) 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 \({\omega(x) := \lim_{t\to\infty} h(x + m(t) ,t)}\) exists and is non-trivial. In particular, as in Bramson’s results for the non-linear FKPP equation, we recover the celebrated \({-3/2\log t}\) correction for initial conditions decaying faster than \({x^{\nu}e^{-x}}\) for some \({\nu < -2}\). 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 \({\omega(x)}\). When h 0(x) decays faster than \({x^{\nu}e^{-x}}\) for some \({\nu < -3}\), we show that r(t) must be chosen to be \({-3\sqrt{\pi/t}}\), 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 \({x^{\nu}e^{-x}}\) for some \({\nu \in [-3, -2)}\), we show that even though we are still in the regime where Bramson’s correction is \({-3/2 \log t}\), 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.
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Berestycki, J., Brunet, É., Harris, S.C. et al. Vanishing Corrections for the Position in a Linear Model of FKPP Fronts. Commun. Math. Phys. 349, 857–893 (2017). https://doi.org/10.1007/s00220-016-2790-9
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DOI: https://doi.org/10.1007/s00220-016-2790-9