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A heat equation with memory: large-time behavior. (English) Zbl 1472.35049

Summary: 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 \(\mathbb{R}^N\) (\(0 < \alpha < 1\)). 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, \(1 \leq p \leq \infty \), depends greatly on the space-time scale under consideration. This result explains in particular the so called “critical dimension phenomenon” (cf. [J. Kemppainen et al., Math. Ann. 366, No. 3–4, 941–979 (2016; Zbl 1354.35178)]).
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.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35A08 Fundamental solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35R11 Fractional partial differential equations

Citations:

Zbl 1354.35178
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References:

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