Well-Posedness and Asymptotic Estimate for a Diffusion Equation with Time-Fractional Derivative

Abstract

In this paper, the authors study the well-posedness and the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term t^−α as t goes to infinity.

Keywords

Mixed-order fractional diffusion equation / Initial-boundary value problem / Asymptotic estimate / Energy method / 35R11 / 35B40 / 26A33 / 34A08 / 35B50

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