Communications in Mathematical Sciences

Volume 19 (2021)

Number 2

Large time behavior for a Hamilton–Jacobi equation in a critical coagulation-fragmentation model

Pages: 495 – 512



Hiroyoshi Mitake (Graduate School of Mathematical Sciences, University of Tokyo, Japan)

Hung V. Tran (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Truong-Son Van (Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)


We study the large-time behavior of the sublinear viscosity solution to a singular Hamilton–Jacobi equation that appears in a critical coagulation-fragmentation model with multiplicative coagulation and constant fragmentation kernels. Our results include complete characterizations of stationary solutions and optimal conditions to guarantee large-time convergence. In particular, we obtain convergence results under certain natural conditions on the initial data, and a nonconvergence result when such conditions fail.


critical coagulation-fragmentation equations, singular Hamilton–Jacobi equations, Bernstein transform, large-time behaviors, nonconvergence results, viscosity solutions

2010 Mathematics Subject Classification

35B40, 35D40, 35F21, 44A10, 45J05, 49L20, 49L25

Received 7 May 2020

Accepted 14 September 2020

Published 12 April 2021