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On initial and terminal value problems for fractional nonclassical diffusion equations


Authors: Nguyen Huy Tuan and Tomás Caraballo
Journal: Proc. Amer. Math. Soc. 149 (2021), 143-161
MSC (2010): Primary 26A33, 35B65, 35R11
DOI: https://doi.org/10.1090/proc/15131
Published electronically: June 11, 2020
Uncorrected version: Original version posted June 11, 2020
Corrected version: This paper was updated due to incomplete funding information at the time of electronic publication.
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Abstract: 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
Email: nhtuan@hcmus.edu.vn

Tomás Caraballo
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico C/ Tarfia s/n, Facultad de Matemáticas, Universidad de Sevilla, Sevilla 41012, Spain
Email: caraball@us.es

DOI: https://doi.org/10.1090/proc/15131
Keywords: Fractional nonclassical diffusion equation, well-posedness, ill-posedness, regularity estimates, regularization and error estimate.
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
Article copyright: © Copyright 2020 American Mathematical Society