Abstract In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei [11]. First, the existence, the positivity and the long time behavior of solutions of the diffusion equation without source term are established by using the Fourier analysis technique. Then, based on the representation of the solution of the inhomogenous linear ordinary differential equation with the general Caputo-type operator, the general diffusion equation with source term is studied.

中文翻译：

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一般时间分数阶扩散方程的柯西问题

摘要 在目前的工作中，我们考虑了 Kochubei [11] 提出的涉及一般 Caputo 型微分算子的时间分数扩散方程的柯西问题。首先，利用傅立叶分析技术建立了无源项扩散方程解的存在性、正性和长时间行为。然后，基于非齐次线性常微分方程解的一般Caputo型算子的表示，研究了带源项的一般扩散方程。