We analyze the one-dimensional pressureless Euler–Poisson equations with linear damping and nonlocal interaction forces. These equations are relevant for modeling collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic behavior of classical solutions with the initial data in the subcritical region.
Bibliographical noteKAUST Repository Item: Exported on 2022-06-02
Acknowledgements: J.A.C. was partially supported by the Royal Society via a Wolfson Research MeritAward. Y.P.C. was supported by the ERC-Starting Grant HDSPCONTR “High-Dimensional Sparse Optimal Control”. J.A.C. and Y.P.C. were partially supportedby EPSRC Grant EP/K008404/1. E.Z. has been partly supported by the NationalScience Center Grant 2014/14/M/ST1/00108 (Harmonia). The authors warmlythank Sergio P ́erez for providing us with the numerical results included in Sec.4.We also thank the Department of Mathematics at KAUST, and particularly A.Tzavaras, for their hospitality during part of this work.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.