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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J. Kemppainen, J. Siljander and R. Zacher were partially supported by Academy of Finland project 138738 (Nonlinear PDEs, project lead by professor Juha Kinnunen). J. Siljander was supported by Academy of Finland postdoctoral research Grant 259363. V. Vergara was partially supported by FONDECYT Grant 1150230. R. Zacher was supported by a Heisenberg fellowship of the German Research Foundation (DFG), GZ Za 547/3-1.
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Kemppainen, J., Siljander, J., Vergara, V. et al. Decay estimates for time-fractional and other non-local in time subdiffusion equations in \(\mathbb {R}^d\) . Math. Ann. 366, 941–979 (2016). https://doi.org/10.1007/s00208-015-1356-z
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DOI: https://doi.org/10.1007/s00208-015-1356-z