We consider a nonlinear heat equation with a nonlinear term
depending on the unknown and its gradient. In the earlier literature,
we have the existence of a stable blow-up solution with an explicit
profile. By pursuing a limited development, we are trapped in scales of
order $\frac{1}{\log |T − t|^\nu}$ where $T$ is the blow-up time. In
this work, we are able to provide a sharper description up to $|T −
t|^\mu$. The price to pay is to replace the explicit profile by a non
explicit profile which is in a fact a well prepared blow-up solution of
the semilinear heat equation with a pure power non linearity.