Global Existence and Sharp Decay Estimates of Classical Solutions to the Vlasov-Poisson System with Radiation Damping

In this paper we consider the two-species Vlasov-Poisson system with a radiation damping term \(D^{[3]}(t)\) in the whole space \(\mathbb {R}^3\), which was introduced by Bauer [Kinet. Relat. Models 11 (2018), 25–42] to approximate the relativistic Vlasov-Maxwell system, a fundamental model of dynamics of collisionless plasma. We obtain the global existence of solutions and optimal pointwise decay estimates of the charge densities and the electrostatic potential to this system for small initial data without any compact support assumptions. To prove our results, we mainly use the modified vector field method and a bootstrap method. There are two main novelties in our arguments: we introduce new modified functions of modified vector fields to control the troublesome terms involving \(D^{[3]}(t)\) since it leads to loss an order derivative, and we raise a new bootstrap assumption and carry out new bootstrap arguments.