On the maximum principle for a time-fractional diffusion equation
On the maximum principle for a time-fractional diffusion equation
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DOI：
10.1515/fca-2017-0060
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发表时间：
2017-01
影响因子：
3
通讯作者：
Yuri Luchko;Masahiro Yamamoto
中科院分区：
文献类型：
--
作者：
Yuri Luchko;Masahiro Yamamoto
本文讨论了在齐次Dirichlet边界条件下，∂tαu(x,t)=∑i,j=1n∂i(aij(x)∂ju(x,t))+c(x)u(x,t)+F(x,t),t>0,x∈Ω∧Rn, $$\begin{array}{} \displaystyle \partial_t^{\alpha} u(x,t) = \sum\limits_{i,j=1}^n \partial_i(a_{ij}(x)\partial_j u(x,t))+ c(x)u(x,t) + F(x,t),\,\, t\gt 0,\,\,x \in \Omega \subset {\mathbb R}^n, \end{array} $$与阶α∈(0,1)的Caputo时间导数的极大值原理。与已经发表的结果相比，我们的发现有两个重要的特点。首先，我们推导出分数Sobolev空间中适当定义的弱解的极大值原理，而不是强解的极大值原理。其次，对于非负源函数F = F(x, t)，利用空间微分算子的零阶导数，在不限制系数c = c(x)符号的情况下，证明了所考虑问题弱解的非负性。此外，我们还证明了解对系数c = c(x)的单调性。
Abstract In this paper, we discuss the maximum principle for a time-fractional diffusion equation ∂tαu(x,t)=∑i,j=1n∂i(aij(x)∂ju(x,t))+c(x)u(x,t)+F(x,t),t>0,x∈Ω⊂Rn,$$\begin{array}{} \displaystyle \partial_t^{\alpha} u(x,t) = \sum\limits_{i,j=1}^n \partial_i(a_{ij}(x)\partial_j u(x,t))+ c(x)u(x,t) + F(x,t),\,\, t\gt 0,\,\,x \in \Omega \subset {\mathbb R}^n, \end{array} $$ with the Caputo time-derivative of the order α ∈ (0, 1) in the case of the homogeneous Dirichlet boundary condition. Compared to the already published results, our findings have two important special features. First, we derive a maximum principle for a suitably defined weak solution in the fractional Sobolev spaces, not for the strong solution. Second, for the non-negative source functions F = F(x, t) we prove the non-negativity of the weak solution to the problem under consideration without any restrictions on the sign of the coefficient c = c(x) by the derivative of order zero in the spatial differential operator. Moreover, we prove the monotonicity of the solution with respect to the coefficient c = c(x).