Non-uniqueness for the forced one-dimensional Vlasov-Poisson equation

We prove non-uniqueness for the forced one-dimensional Vlasov-Poisson equation on the whole space. More precisely, for every finite $p$, we construct a nonnegative ion density, smooth for positive times, and a source $S\in C([0,1];L^1({\mathbb{R}}^2))$ with $S(0)=0$, for which there exist two distinct nonnegative distributional solutions $f_{\pm }\in C([0,1];L^1({\mathbb{R}}^2))$ with the same zero initial datum and with spatial densities bounded in $L^p(\mathbb{R})$ uniformly in time. The result is sharp in terms of the integrability of the spatial densities: our solutions have Lipschitz electric fields, and we show that in this class uniqueness holds as soon as the spatial density is bounded.