  In this paper, we deal with models with Born-Infeld type diffusion and
monostable reaction, investigating the effect of the introduction of a
convection term on the limit shape of the critical front profile for vanishing
diffusion. We first provide an estimate of the critical speed and then, through
a careful analysis of an equivalent first-order problem, we show that different
convection terms may lead either to a complete sharpening of the limit profile
or to its complete regularization, presenting some related numerical
simulations.
