On initial and terminal value problems for fractional nonclassical diffusion equations

Tuan, Nguyen Huy; Caraballo, Tomás

doi:10.1090/proc/15131

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 norm is first established.

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