arXiv,
CIMP,
pdf.
Take the linearised FKPP equation
∂
_{t} h=∂
_{xx}
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
ω(
x):=lim
_{t} 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^{ν}e
^{−x}
for some ν<−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). When
h_{0}(
x) decays faster than
x^{ν}e
^{−x}
for some ν<−3, we show
that
r(
t) must be chosen to be −3(π/
t)
^{1/2} which is precisely the term predicted
heuristically by Ebert-van Saarloos in the non-linear case. When the initial
condition decays as
x^{ν}e
^{−x}
for some ν∈[−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 using an analytical approach and only for compactly supported
initial conditions.