Pointwise gradient estimates and stabilization for Fisher-KPP type equations with a concentration dependent di¤usion

We prove a pointwise gradient estimate for the bounded weak solution of the Cauchy problem associated to the quasilinear Fisher-KPP type equation ut = '(u)xx + (u) when ' satis…es that '(0)=0; and (u) is vanishing only for levels u = 0 and u = 1. As a …rst consequence we prove that the bounded weak solution becomes instantaneously a continuous function even if the initial datum is merely a discontinuous bounded function. Moreover the obtained estimates also prove the stabilization of the gradient of bounded weak solutions as t ! +1 for suitable initial data.