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Wellposedness and smoothing properties of history-state-based variable-order time-fractional diffusion equations

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Abstract

We prove the wellposedness of history-state-based variable-order linear time-fractional diffusion equations in multiple space dimensions. We also prove that the regularity of their solutions depends on the behavior of the variable order at the initial time \(t=0\), in addition to the usual smoothness assumptions. More precisely, we prove that their solutions have full regularity (i.e., the solutions can achieve high-order smoothness under high-order regularity assumptions of the data) as their integer-order analogs if the variable order has an integer limit at \(t=0\) or exhibits singular behaviors at \(t=0\) like in the case of the constant-order time-fractional diffusion equations if the variable order has a non-integer value at \(t=0\).

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Acknowledgements

The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.

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  1. University of South Carolina, 1523 Greene St., Columbia, SC, 29208, USA

    Xiangcheng Zheng & Hong Wang

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  1. Xiangcheng Zheng
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  2. Hong Wang
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Correspondence to Hong Wang.

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This work was funded by the OSD/ARO MURI Grant W911NF-15-1-0562 and by the National Science Foundation under Grant DMS-1620194.

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Zheng, X., Wang, H. Wellposedness and smoothing properties of history-state-based variable-order time-fractional diffusion equations. Z. Angew. Math. Phys. 71, 34 (2020). https://doi.org/10.1007/s00033-020-1253-5

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  • DOI: https://doi.org/10.1007/s00033-020-1253-5

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