We investigate a parabolic-elliptic system for maps from a compact Riemann surface M into a Lorentzian manifold N × ℝ with a warped product metric. We prove a global existence result of the parabolic-elliptic system by assuming either some geometric conditions on the target Lorentzian manifold or small energy of the initial maps. The result implies the existence of a harmonic map in a given homotopy class with fixed boundary data.