The well-posedness and regularity of mild solutions to the time-fractional Cable equation

This paper investigates the well-posedness of mild solutions for a linear time-fractional Cable equation on a bounded domain $ \Omega \subset \mathbb{R}^d $ ($ d \geq 1 $) with a $ C^2 $ boundary:

$ \begin{align*} \left\{ \begin{aligned} &\partial_t u = \partial_t^{1-\alpha}\Delta u - \partial_t^{1-\beta}u + f, \quad (t, x) \in (0, T) \times \Omega, \\ &u(0, x) = u_0(x), \quad x\in \Omega, \\ &u = 0, \quad x\in\partial\Omega, \end{aligned} \right. \end{align*} $

where $ 0 < \alpha $, $ \beta < 1 $, and $ \partial_{t}^{1-\beta} $ and $ \partial_{t}^{1-\alpha} $ denote the Riemann–Liouville fractional derivatives of orders $ 1-\beta $ and $ 1-\alpha $, 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.