We prove existence of classical solutions to the so-called diffusive Vesicle Supply Centre (VSC) model describing the growth of fungal hyphae. It is supposed in this model that the local expansion of the cell wall is caused by a flux of vesicles into the wall and that the cell wall particles move orthogonally to the cell surface. The vesicles are assumed to emerge from a single point inside the cell (the VSC) and to move by diffusion. For this model, we derive a non-linear, non-local evolution equation and show the existence of solutions relevant to our application context, namely, axially symmetric surfaces of fixed shape, travelling along with the VSC at constant speed. Technically, the proof is based on the Schauder fixed point theorem applied to Hölder spaces of functions. The necessary estimates rely on comparison and regularity arguments from elliptic PDE theory
