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

Abstract

Abstract 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.

Keywords

• Fractional diffusion equation
• Mittag-Leffler function
• M-Wright function
• Mikhlin multiplier theorem
• Weighted Sobolev space
• Existence