Variable-exponent fractional models attract increasing attentions in various applications, while the rigorous analysis is far from well developed. This work provides general tools to address these models. Specifically, we first develop a convolution method to study the well-posedness, regularity, an inverse problem and numerical approximation for the sundiffusion of variable exponent. For models such as the variable-exponent two-sided space-fractional boundary value problem (including the variable-exponent fractional Laplacian equation as a special case) and the distributed variable-exponent model, for which the convolution method does not apply, we develop a perturbation method to prove their well-posedness. The relation between the convolution method and the perturbation method is discussed, and we further apply the latter to prove the well-posedness of the variable-exponent Abel integral equation and discuss the constraint on the data under different initial values of variable exponent.

翻译：变指数分数阶模型在各种应用中日益受到关注，但其严格分析远未成熟。本文提供了处理这些模型的一般性工具。具体而言，我们首先发展了一种卷积方法，用于研究变指数次扩散的适定性、正则性、反问题及数值逼近。对于卷积方法不适用的情况，例如变指数双侧空间分数阶边值问题（包括变指数分数阶Laplacian方程作为特例）和分布式变指数模型，我们发展了一种扰动方法以证明其适定性。本文讨论了卷积方法与扰动方法之间的关系，并进一步应用扰动方法证明了变指数Abel积分方程的适定性，同时讨论了变指数不同初值条件下数据的约束条件。