We study the large-time behavior in all $L^p$ norms of solutions to a heat equation with a Caputo α-time derivative posed in ${\mathbb{R}}^N$ ($0<\alpha <1$). These are known as subdiffusion equations. The initial data are assumed to be integrable, and, when required, to be also in $L^p$.

We find that the decay rate in all $L^p$ norms, $1\le p\le \infty$, depends greatly on the space-time scale under consideration. This result explains in particular the so called “critical dimension phenomenon” (cf. [21]).

Moreover, we find the final profiles (that strongly depend on the scale). The most striking result states that in compact sets the final profile (in all $L^p$ norms) is a multiple of the Newtonian potential of the initial datum.

Our results are very different from the ones for classical diffusion equations and show that, in accordance with the physics they have been proposed for, these are good models for particle systems with sticking and trapping phenomena or fluids with memory.

我们研究所有的长时间行为 ${升}^{磷}$具有 Caputo α时间导数的热方程解的范数${\mathbb{电阻}}^N$ ($0<\alpha <1$）。这些被称为子扩散方程。初始数据被假定为可积的，并且在需要时，也在${升}^{磷}$.

我们发现所有的衰减率 ${升}^{磷}$ 规范， $1\le 磷\le \infty$，很大程度上取决于所考虑的时空尺度。这个结果特别解释了所谓的“临界维度现象”（参见 [21]）。

此外，我们找到了最终的配置文件（在很大程度上取决于规模）。最引人注目的结果表明，在紧凑型设置中最终配置文件（在所有${升}^{磷}$ 范数）是初始数据的牛顿势的倍数。

我们的结果与经典扩散方程的结果非常不同，并表明，根据它们提出的物理学，这些是具有粘附和捕获现象的粒子系统或具有记忆的流体的良好模型。