## 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

## 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.## References

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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: 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
- ORCID: 0000-0003-4697-898X
- Email: caraball@us.es
- 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: https://doi.org/10.1090/proc/15131
- MathSciNet review: 4172593