In this talk, we consider some specific reaction-diffusion equations in order to understand the equivalence between asymptotic Turing instability of a steady state and backward property of some parabolic equations or cross-diffusion equations in the formal limit of fat reaction terms. We will see that the structure of the studied equations involves some Lyapunov functions which leads to a priori estimates allowing to pass rigorously for the fast reaction terms in the case without Turing instabilities.