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In this paper we consider the Cauchy problem of a generalization of time-fractional diffusion equation with variable coefficients in R+n+1$\mathbb {R}_{+}^{n+1}$, where the time derivative is replaced by a regularized hyper-Bessel operator. The explicit solution of the inhomogeneous linear equation for any n∈Z+$n\in\mathbb {Z}^{+}$ and its uniqueness in a weighted Sobolev space are established. The key tools are Mittag-Leffler functions, M-Wright functions and Mikhlin multiplier theorem. At last, we obtain the existence of solution of the semilinear equation for n=1$n=1$ by using a fixed point theorem.

Existence results for a generalization of the time-fractional diffusion equation with variable coefficients