  In a recent paper, we have introduced a new model to describe front
propagation in bushfires. This model describes temperature diffusion in view of
an ignition process induced by an interaction kernel, the effect of the
environmental wind and that of the fire wind.
  This has led to the introduction of a new partial differential equation of
evolutionary type, with nonlinear terms both of integral kind and involving the
gradient of the solution. This equation is, in a sense, ``hybrid'', since it
encodes both analytical and geometric features of the front propagation. This
new characteristic makes the equation particularly interesting also from the
mathematical point of view, often falling outside the territory already covered
by standard methods.
  In this paper, we start the mathematical treatment of this equation by
establishing the short time and global existence theory for this equation.
