The dynamics of the growth of interfaces in the presence of noise and when
the normal velocity is constant, in the weakly nonlinear limit, are describ
ed by the Kardar-Parisi-Zhang (KPZ) equation. In many applications, however
, the growth is controlled by nonlocal transport, which is not contained in
the original KPZ equation. For these problems we are proposing an extensio
n of the KPZ model, where the nonlocal contribution is expressed through a
Hilbert transform and can act to either stabilize or destabilize the interf
ace. The model is illustrated with a specific example from reactive infiltr
ation. The properties of the solution of the resulting equation are studied
in one spatial dimension in the linear and the nonlinear limits, for both
stable and unstable growth. We find that the early-time behavior has a powe
r-law scaling similar to that of the KPZ equation. However, in the case of
stable growth, the scaling of the. saturation width is logarithmic, which d
iffers from the power law in the KPZ equation. This dependence reflects the
stabilizing effect of nonlocal transport. Tn the unstable case, we obtain
results similar to those of Olami et al. [Phys. Rev. E 55, 2649 (1997)].