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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On initial and terminal value problems for fractional nonclassical diffusion equations
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by Nguyen Huy Tuan and Tomás Caraballo PDF
Proc. Amer. Math. Soc. 149 (2021), 143-161 Request permission


In this paper, we consider fractional nonclassical diffusion equations under two forms: initial value problem and terminal value problem. For an initial value problem, we study local existence, uniqueness, and continuous dependence of the mild solution. We also present a result on unique continuation and a blow-up alternative for mild solutions of fractional pseudo-parabolic equations. For the terminal value problem, we show the well-posedness of our problem in the case $0<\alpha \le 1$ and show the ill-posedness in the sense of Hadamard in the case $\alpha > 1$. Then, under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic-type in $L^q$ norm is first established.
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Additional Information
  • Nguyen Huy Tuan
  • Affiliation: Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam; and Vietnam National University, Ho Chi Minh City, Vietnam
  • MR Author ID: 777405
  • ORCID: 0000-0002-6962-1898
  • Email:
  • Tomás Caraballo
  • Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla, Sevilla 41012, Spain
  • ORCID: 0000-0003-4697-898X
  • Email:
  • Received by editor(s): November 7, 2019
  • Received by editor(s) in revised form: January 15, 2020
  • Published electronically: June 11, 2020
  • Additional Notes: This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2019.09
    The research of the second author was partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00.
  • Communicated by: Wenxian Shen
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 149 (2021), 143-161
  • MSC (2010): Primary 26A33, 35B65, 35R11
  • DOI:
  • MathSciNet review: 4172593