Motivated by the industrial manufacture of organic light-emitting-diode displays, we formulate and analyse a mathematical model for the evolution of a thin droplet in a shallow axisymmetric well of rather general shape both before and after touchdown that accounts for the spatially non-uniform evaporation of the fluid, perform physical experiments using three cylindrical wells with different small aspect ratios, and validate the mathematical model by comparing the present experimental results with the corresponding theoretical predictions for a cylindrical well.