Decay estimates for time-fractional and other non-local in time subdiffusion equations in R^d
Kemppainen, J., Siljander, J., Vergara, V., & Zacher, R. (2016). Decay estimates for time-fractional and other non-local in time subdiffusion equations in R^d. Mathematische Annalen, 366(3), 941-979. https://doi.org/10.1007/s00208-015-1356-z
Published inMathematische Annalen
© Springer-Verlag Berlin Heidelberg 2016. This is a final draft version of an article whose final and definitive form has been published by Springer. Published in this repository with the kind permission of the publisher.
We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in R d . An important special case is the timefractional 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.