Asymptotic behavior of solutions for nonlinear 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 M m ( Ω ) , then ‖ u ( t ) ‖ L r ( Ω ) ∗ ≤ C ‖ u 0 ‖ L m ( Ω ) ∗ t N 2 1 m - 1 r , ∀ t > 0 , thus extending to Marcinkiewicz spaces the results which hold for data in Lebesgue spaces.