  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.
