Backward stochastic differential equations with random stopping time and singular final condition

In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: $$Y_t=\xi -\int_{t\wedge \tau}^{\tau}Y_r|Y_r|^q dr-\int_{t\wedge \tau}^{\tau}Z_r dB_r,\qquad t\geq 0,$$ where $\tau$ is a stopping time, $q$ is a positive constant and $\xi$ is a $\mathcal{F}_{\tau}$-measurable random variable such that $\mathbf{P}(\xi =+\infty)>0$. We study the link between these BSDE and the Dirichlet problem on a domain $D\subset \mathbb{R}^d$ and with boundary condition $g$, with $g=+\infty$ on a set of positive Lebesgue measure. We also extend our results for more general BSDE.