  The paper is concerned with a zero-sum differential game in the case where a
payoff is determined by the exit time, that is, the first time when the system
leaves the game domain. Additionally, we assume that a part of domain's
boundary is a lifeline where the payoff is infinite. Hereby, the examined
problem generalizes the well-known time-optimal problem as well as time-optimal
problem with lifeline. The main result of the paper relies on the solution to
the Direchlet problem for the Hamilton-Jacobi equation associated with the game
with exit time. We prove the existence of the value function for examined
problem and construct suboptimal feedback strategies under assumption that the
associated Dirichlet problem for the Hamilton-Jacobi equation admits a
viscosity/minimax solution. Additionally, we derive a sufficient condition of
existence result to this Dirichlet problem.
