Contents 
It is well known that the solution of the heat equation
with summable initial datum $u_0$ becomes "immediately bounded" and satisfies the decay estimate
\begin{equation}\label{eqca}
\ u(\cdot,t)\_{L^\infty} \leq C \frac{\u_0\_{L^{1}}}{t^{\frac{N}{2}}}, \quad t> 0.
\end{equation}
This "strong regularizing effect" is not a peculiarity of the heat equation since it appears also for a lot of other parabolic problems, also nonlinear, degenerate or singular like the pLaplacian equation, the porous medium equation, the fast diffusion equation, etc. In a recent paper we have proved that this phenomenon occurs when the solutions satisfy certain integral inequalities.
We investigate here what happens when this regularizing effect does not appear and which is the solutions' behavior in this case. 
