Asymptotic behavior for some parabolic problems with Marcinkiewicz data

In this paper we prove the asymptotic behavior, as $t$ tends to zero, of solutions of nonlinear parabolic equations with initial data belonging to Marcinkiewicz spaces. Namely, that if the initial datum $u_{0}$ belongs to $\emme{m}$, then $$ \marc{u(t)}{\elle{r}} \leq \mathcal{C}\,\frac{\marc{u_{0}}{\elle{m}}}{t^{\frac{N}{2}\left( \frac{1}{m} - \frac{1}{r}\right)}}\,, \qquad \forall\,t > 0\,, $$ thus extending to Marcinkiewicz spaces the results which hold for data in Lebesgue spaces.