Coulomb-bound electron-hole pairs, or excitons, have been in the focus of the solid-state research for many decades. They are of paramount importance for the fundamental understanding of interacting charge carriers in semiconductors. Recently, excitons in single layers of semiconducting transition-metal dichalcogenides (TMDCs) were found to combine a nnumber of intriguing properties, including binding energies on the order of 0.5 eV, strong light-matter interaction and spin-valley coupling.

Here, we address the topic of exciton transport in TMDCs by directly monitoring spatial behavior of excitons in freestanding and supported single layers of WS₂ through spatially- and time-resolved photoluminescence. We find highly nonlinear behavior with characteristic, qualitative changes in the spatial profiles of the exciton emission and an effective diffusion coefficient increasing from 0.3 cm²/s to more than 30 cm²/s, depending on the injected exciton density. Solving the diffusion equation while accounting for Auger recombination allows us to identify the main origin of the nonlinearity. At elevated excitation densities, the excitons distribution evolves into long-lived halo shapes with micrometer-scale diameter, indicating additional memory effects in the dynamics.