  We consider a class of aggregation-diffusion equations on unbounded one
dimensional domains with Lipschitz nonincreasing mobility function. We show
strong $L^1$-convergence of a suitable deterministic particle approximation to
weak solutions of a class aggregation-diffusion PDEs (coinciding with the
classical ones in the no vacuum regions) for any bounded initial data of finite
energy. In order to prove well-posedness and convergence of the scheme with no
BV or no vacuum assumptions and overcome the issues posed in this setting by
the presence of a mobility function, we improve and strengthen the techniques
introduced in arXiv:2012.01966(2).
