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| We prove that any subcritical solution to the Becker-D\"{o}ring
equations converges exponentially fast to the unique steady state
with same mass. Our convergence result is quantitative and we show
that the rate of exponential decay is governed by the spectral gap
for the linearized equation, for which several bounds are
provided. This improves the known convergence result by Jabin \&
Niethammer (2003). Our approach is based on a careful spectral
analysis of the linearized Becker-D\"{o}ring equation (which is new
to our knowledge) in both a Hilbert setting and in certain weighted
$\ell^1$ spaces. This spectral analysis is then combined with
uniform exponential moment bounds of solutions in order to obtain a
convergence result for the nonlinear equation. |
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