GENERAL TIME-FRACTIONAL DIFFUSION EQUATION: SOME UNIQUENESS AND EXISTENCE RESULTS FOR THE INITIAL-BOUNDARY-VALUE PROBLEMS
GENERAL TIME-FRACTIONAL DIFFUSION EQUATION: SOME UNIQUENESS AND EXISTENCE RESULTS FOR THE INITIAL-BOUNDARY-VALUE PROBLEMS
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DOI：
10.1515/fca-2016-0036
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发表时间：
2016-06-01
影响因子：
3
通讯作者：
Yamamoto, Masahiro
中科院分区：
文献类型：
--
作者：
Luchko, Yuri;Yamamoto, Masahiro
本文讨论了一般的时间分数阶扩散方程的初边值问题，推广了单项和多项时间分数阶扩散方程以及分布阶的时间分数阶扩散方程。首先，给出了函数在极大值点的Riemann-Liouville和Caputo型一般时间分数导数的重要估计。应用这些估计证明了一般时间分数阶扩散方程的一个弱极大值原理。作为极大值原理的应用，建立了该方程在Dirichlet边界条件下初边值问题强解和弱解的唯一性。最后证明了具有齐次边界条件的初边值问题的适当定义的广义解的存在性。
In this paper, we deal with the initial-boundary-value problems for a general time-fractional diffusion equation which generalizes the single- and the multi-term time-fractional diffusion equations as well as the time-fractional diffusion equation of the distributed order. First, important estimates for the general time-fractional derivatives of the Riemann-Liouville and the Caputo type of a function at its maximum point are derived. These estimates are applied to prove a weak maximum principle for the general time-fractional diffusion equation. As an application of the maximum principle, the uniqueness of both the strong and the weak solutions to the initial-boundary-value problem for this equation with the Dirichlet boundary conditions is established. Finally, the existence of a suitably defined generalized solution to the the initial-boundary-value problem with the homogeneous boundary conditions is proved.