From Option Valuation under Stochastic Volatility II:

What is it? This chart shows early exercise boundaries for the American-style put under discrete dividend models. The models are associated to Black-Scholes (geometric Brownian motion, or GBM) stock price evolution. How to handle discrete dividends is, remarkably (even in this basic GBM case), still a contentious issue more than 40 years after Black and Scholes’ seminal publication.

Two general approaches are: (i) piecewise GBM, using various dividend policies \mathcal{D}(S) , and (ii) escrowed dividends. The first two rows of the chart show the early exercise boundary in the (t,S) plane under piecewise GBM and various \mathcal{D}(S). The last row shows the same in the (t,\tilde{S}) plane under the escrowed dividend model, where \tilde{S} is the risky part of the stock price. More details are found in Chapter 9 of the latest book.