We consider the solution of a nonlinear Kraichnan equation $$partial_s H(s,t)=int_t^s H(s,u)H(u,t) k(s,u) du,quad sge t$$ with a covariance kernel $k$ and boundary condition $H(t,t)=1$. We study the long time behaviour of $H$ as the time parameters $t,s$ go to infinity, according to the asymptotic behaviour of $k$. This question appears in various subjects since it is related with the analysis of the asymptotic behaviour of the trace of non-commutative processes satisfying a linear differential equation, but also naturally shows up in the study of the so-called response function and aging properties of the dynamics of some disordered spin systems.