Almost global existence and transient self similar decay for Poiseuille flow at criticality for exponentially long times

We consider nonlinear diffusion equations with critical exponent as ∂tu=∂x2u+u3 with x∈R for small initial data in L1∩L∞. It is well known that almost all solutions of this system explode in finite time. However, we make the observation that in terms of the norm of the initial conditions it takes an exponentially long time. Moreover, before explosion the L∞-norm of such solutions becomes exponentially small which makes it almost impossible to observe the instability in experiments. As an application we consider the long time transient self similar decay to unstable Poiseuille flow at criticality for exponentially long times. This, together with a subcritical bifurcation and short time transient amplification, is a principal obstruction in all attempts to measure the critical Reynolds number for this experiment more and more precisely.