A heat equation with memory: Large-time behavior

Abstract

© 2021We study the large-time behavior in all Lp norms of solutions to a heat equation with a Caputo α-time derivative posed in RN (0<α<1). These are known as subdiffusion equations. The initial data are assumed to be integrable, and, when required, to be also in Lp. We find that the decay rate in all Lp norms, 1≤p≤∞, 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 Lp 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.