A heat equation with memory: Large-time behavior

We study the large-time behavior in all L-p norms of solutions to a heat equation with a Caputo alpha-time derivative posed in 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 <= p <= infinity, 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 Lpnorms) 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. (C) 2021 Published by Elsevier Inc.

Automated summary

The study focuses on the large-time behavior in all L-p norms of solutions to a heat equation with a Caputo alpha-time derivative, specifically for subdiffusion equations and integrable initial data. The decay rate in all L-p norms is found to greatly depend on the space-time scale, explaining the critical dimension phenomenon. Additionally, the final profiles strongly depend on scale, with the final profile in compact sets being a multiple of the Newtonian potential of the initial datum. These results differ significantly from classical diffusion equations and indicate that these equations are good models for systems with sticking and trapping phenomena or fluids with memory.