Asymptotic behavior of solutions for nonlinear parabolic problems with Marcinkiewicz data

Abstract

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 $M^{m}(\Omega )$, then

$$\begin{aligned} \Vert u(t)\Vert _{\scriptstyle L^{r}(\Omega )}^{*} \le {\mathcal {C}}\,\frac{\Vert u_{0}\Vert _{\scriptstyle L^{m}(\Omega )}^{*}}{t^{\frac{N}{2}\left( \frac{1}{m} - \frac{1}{r}\right) }}, \qquad \forall \,t > 0, \end{aligned}$$

thus extending to Marcinkiewicz spaces the results which hold for data in Lebesgue spaces.