Kemppainen, J., Siljander, J., Vergara, V. et al. Math. Ann. (2016) 366: 941. https://doi.org/10.1007/s00208-015-1356-z

### Decay estimates for time-fractional and other non-local in time subdiffusion equations in $$\mathbb {R}^d$$

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Author: Kemppainen, Jukka1; Siljander, Juhana2; Vergara, Vicente3,4;
Organizations: 1Department of Mathematical Sciences, University of Oulu, Pentti Kaiteran katu 1, PO Box 3000, 90014 Oulu, Finland
2Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland
3Departamento de Matemáticas, Universidad de La Serena, La Serena, Chile
4Dirección General de Investigación, Universidad de Tarapacá, 1520 Arica, Chile
5Institute of Applied Analysis, Ulm University, 89069 Ulm, Germany
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.5 MB)
Language: English
Published: Springer Nature, 2016
Publish Date: 2020-02-07
Description:

# Abstract

We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in $$\mathbb {R}^d$$. An important special case is the time-fractional diffusion equation, which has seen much interest during the last years, mostly due to its applications in the modeling of anomalous diffusion processes. We follow three different approaches and techniques to study this particular case: (A) estimates based on the fundamental solution and Young’s inequality, (B) Fourier multiplier methods, and (C) the energy method. It turns out that the decay behaviour is markedly different from the heat equation case, in particular there occurs a critical dimension phenomenon. The general subdiffusion case is treated by method (B) and relies on a careful estimation of the underlying relaxation function. Several examples of kernels, including the ultraslow diffusion case, illustrate our results.

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Series: Mathematische Annalen
ISSN: 0025-5831
ISSN-E: 1432-1807
ISSN-L: 0025-5831
Volume: 366
Issue: 3-4
Pages: 941 - 979
DOI: 10.1007/s00208-015-1356-z