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Abstract
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 nontrivial. In particular, as in Bramson’s results for the nonlinear 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 nonlinear 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 120, 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.
Original language  English 

Pages (fromto)  857893 
Number of pages  37 
Journal  Communications in Mathematical Physics 
Volume  349 
Issue number  3 
Early online date  23 Dec 2016 
DOIs  
Publication status  Published  30 Jan 2017 
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Projects
 1 Finished

EPSRC Posdoctoral Fellowship in Applied Probability for Dr Matthew I Roberts
Engineering and Physical Sciences Research Council
3/04/13 → 2/07/16
Project: Research council
Profiles

Matthew Roberts
 Department of Mathematical Sciences  Royal Society University Research Fellow & Reader
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
Person: Research & Teaching