  We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact
Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a
warped product metric. That system turns the harmonic map type equations into a
parabolic system, but keeps the $v$-equation as a nonlinear second order
constraint along the flow. 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 Lorentzian harmonic map in a given homotopy class
with fixed boundary data.
