Before revision:  In this paper we prove that when the dividend function is positive and concave, then the boundary  tends to 0 as time tends to t_d^- and is non-increasing in a left-hand neighbourhood of t_d. We also show that the exercise boundary is continuous and a high contact principle holds in such a neighbourhood when the dividend function is moreover linearin a neighbourhood of 0.
After revision:  In this paper we prove that when the dividend function is positive and concave, then the boundary tends to 0 as time tends to t_d^- with a speed that we can characterize. When the dividend function is linear in a neighbourhood of zero, then we show continuity of the exercise boundary is continuous and a high contact principle holds in such a neighbourhood when the dividend function is moreover linearin a neighbourhood of 0.