@article{Tan2025, 
author = {Hujing Tan and Pu Wang},
title = {The well-posedness and regularity of mild solutions to the time-fractional Cable equation},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {7},
pages = {16624-16641},
keywords = {time-fractional Cable equation, well-posedness, weighted Hölder space},
url = {https://www.sciopen.com/article/10.3934/math.2025745},
doi = {10.3934/math.2025745},
abstract = {This paper investigates the well-posedness of mild solutions for a linear time-fractional Cable equation on a bounded domain    Ω  ⊂            R        d   (   d  ≥  1) with a        C    2   boundary:                                 {                                                                                    ∂                  t                                u                =                                  ∂                  t                                      1                    −                    α                                                  Δ                u                −                                  ∂                  t                                      1                    −                    β                                                  u                +                f                ,                                (                t                ,                x                )                ∈                (                0                ,                T                )                ×                Ω                ,                                                                                  u                (                0                ,                x                )                =                                  u                  0                                (                x                )                ,                                x                ∈                Ω                ,                                                                                  u                =                0                ,                                x                ∈                ∂                Ω                ,                                                                  where    0  <  α,    β  <  1, and        ∂          t              1      −      β       and        ∂          t              1      −      α       denote the Riemann–Liouville fractional derivatives of orders    1  −  β and    1  −  α, respectively. By employing the eigenfunction expansion method, we constructed the mild solution and established its definition. Utilizing the Banach contraction mapping principle and properties of the Mittag-Leffler function, we derived the existence, uniqueness, and regularity of mild solutions for the linear problem. Furthermore, we introduced a weighted Hölder continuous function space and demonstrated the existence and uniqueness of mild solutions within this frameworks. The results obtained in this work contribute to the theoretical understanding of time-fractional Cable equations and serve as a foundation for further studies in fractional-order diffusion processes.}
}