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The well-posedness analysis of distributed order fractional diffusion problems on \({\mathbb {R}}^N\)

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Abstract

This paper is devoted to the study of a semilinear diffusion problem with distributed order fractional derivatives on \({\mathbb {R}}^N\), which can be used to characterize the ultra-slow diffusion processes with time-dependent logarithmical-law attenuation. We use the resolvents approach to present the local well-posedness of mild solutions belonging to \(L^r(\mathbb {R}^N)\,(r>2)\), in which the \(L^p-L^q\) estimates and continuity of the operator are first established. Then, under the assumption on the initial value belonging to \(L^p({\mathbb {R}}^N)\), the global well-posedness of mild solutions is derived. Moreover, a decay estimate in \(L^r\) norm is included.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (12001462, 12071396) and the Natural Science Foundation of Hunan Province (2020JJ5529).

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Authors and Affiliations

  1. Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, China

    Li Peng & Yong Zhou

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan, 411105, China

    Li Peng

  3. Faculty of Information Technology, Macau University of Science and Technology, Macau, 999078, China

    Yong Zhou

  4. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, China

    Jia Wei He

Authors
  1. Li Peng
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  2. Yong Zhou
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  3. Jia Wei He
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Correspondence to Yong Zhou.

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Communicated by Ansgar Jüngel.

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Peng, L., Zhou, Y. & He, J.W. The well-posedness analysis of distributed order fractional diffusion problems on \({\mathbb {R}}^N\). Monatsh Math 198, 445–463 (2022). https://doi.org/10.1007/s00605-021-01631-8

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  • DOI: https://doi.org/10.1007/s00605-021-01631-8

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