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.

中文翻译：

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Marcinkiewicz 数据非线性抛物线问题解的渐近行为

在本文中，我们证明了初始数据属于 Marcinkiewicz 空间的非线性抛物型方程的解在t趋于零时的渐近行为。即，如果初始数据\(u_{0}\)属于\(M^{m}(\Omega )\)，则

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

因此，将适用于勒贝格空间中的数据的结果扩展到 Marcinkiewicz 空间。