We study timed systems in which some timing features are unknown parameters. Parametric timed automata are a classical formalism for such systems but for which most interesting problems are undecidable. Lower-bound/upper-bound parametric timed automata (L/U-PTAs) achieve decidability for reachability properties by enforcing a separation of parameters used as upper bounds in the automaton constraints, and those used a lower bounds. We further study L/U-PTAs by considering liveness related problems. We prove that: (1) the existence of at least one parameter valuation for which there exists an infinite run in the automaton is PSPACE-complete; (2) the existence of a parameter valuation such that the system has a deadlock is however undecidable; (3) the problem of the existence of a valuation for which a run remains in a given set of locations exhibits a very thin border between decidability and undecidability.