# American Institute of Mathematical Sciences

doi: 10.3934/eect.2020001

## Initial-boundary value problems for multi-term time-fractional diffusion equations with $x$-dependent coefficients

 1 School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255049, China 2 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan 3 Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, no 54, 050094 Bucharest Romania 4 Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

* Corresponding author: Zhiyuan Li

Received  October 2018 Published  August 2019

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with $x$-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

Citation: Zhiyuan Li, Xinchi Huang, Masahiro Yamamoto. Initial-boundary value problems for multi-term time-fractional diffusion equations with $x$-dependent coefficients. Evolution Equations & Control Theory, doi: 10.3934/eect.2020001
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