Well-posedness and asymptotic estimate for a diffusion equation with time-fractional derivative

TL;DR

This paper investigates the well-posedness and long-term decay behavior of solutions to a time-fractional diffusion equation, establishing existence, regularity, and asymptotic decay rates in bounded domains.

Contribution

It provides new results on the existence, regularity, and decay estimates for solutions to a mixed-order time-fractional diffusion equation.

Findings

Unique existence and regularity of solutions established.

Solution decay rate dominated by t^{- extalpha} as t approaches infinity.

Energy method combined with maximum principle used for decay analysis.

Abstract

In this paper, we study 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 $t^{-\alpha}$ as $t\to\infty$.