Take the linearised FKPP equation $${\partial_{t}h = \partial^{2}_{x}h + h}$$ ∂ t h = ∂ x 2 h + h with boundary condition h(m(t), t) = 0. Depending on the behaviour of the initial condition h0(x) = h(x, 0) we obtain the asymptotics—up to a o(1) term r(t)—of the absorbing boundary m(t) such that $${\omega(x) := \lim_{t\to\infty} h(x + m(t) ,t)}$$ 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 $${-3/2\log t}$$ - 3 / 2 log t correction for initial conditions decaying faster than $${x^{\nu}e^{-x}}$$ x ν e - x for some $${\nu < -2}$$ 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 $${\omega(x)}$$ ω ( x ) . When h0(x) decays faster than $${x^{\nu}e^{-x}}$$ x ν e - x for some $${\nu < -3}$$ ν < - 3 , we show that r(t) must be chosen to be $${-3\sqrt{\pi/t}}$$ - 3 π / 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 $${x^{\nu}e^{-x}}$$ x ν e - x for some $${\nu \in [-3, -2)}$$ ν ∈ [ - 3 , - 2 ) , we show that even though we are still in the regime where Bramson’s correction is $${-3/2 \log t}$$ - 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.