Communications in Mathematical Sciences

Volume 15 (2017)

Number 6

Cutoff estimates for the linearized Becker–Döring equations

Pages: 1685 – 1702



Ryan W. Murray (Department of Mathematics, Pennsylvania State University, State College, Penn., U.S.A.)

Robert L. Pego (Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)


This paper continues the authors’ previous study [R. Murray and R. Pego, SIAM J. Math. Anal., 48:2819–2842, 2016] of the trend toward equilibrium of the Becker–Döring equations with subcritical mass, by characterizing certain fine properties of solutions to the linearized equation. In particular, we partially characterize the spectrum of the linearized operator, showing that it contains the entire imaginary axis in polynomially weighted spaces. Moreover, we prove detailed cutoff estimates that establish upper and lower bounds on the lifetime of a class of perturbations to equilibrium.


coagulation-fragmentation equations, spectrum, cutoff estimates

2010 Mathematics Subject Classification

34D05, 47D06, 82C05

This material is based upon work supported by the National Science Foundation under grants DMS 1211161 and DMS 1515400, and partially supported by the Center for Nonlinear Analysis (CNA) under National Science Foundation PIRE Grant no. OISE-0967140, and the NSF Research Network Grant no. RNMS11-07444 (KI-Net).

Received 26 September 2016

Published 27 June 2017