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 $\mathrm{\Omega }\subset {\mathbb{R}}^d$ ( $d\ge 1$) with a $C^2$ boundary:

$$\begin{array}{r}\{\begin{array}{rl}& {\mathrm{\partial }}_{t}u={\mathrm{\partial }}_t^{1-\alpha }\mathrm{\Delta }u-{\mathrm{\partial }}_t^{1-\beta }u+f,(t,x)\in (0,T)\times \mathrm{\Omega },\\ & u(0,x)=u_0(x),x\in \mathrm{\Omega },\\ & u=0,x\in \mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$$

where $0<\alpha$, $\beta <1$, and ${\mathrm{\partial }}_t^{1-\beta }$ and ${\mathrm{\partial }}_t^{1-\alpha }$ 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.