On the maximum principle for a time-fractional diffusion equation

DOI10.1515/FCA-2017-0060zbMATH Open1374.35426arXiv1702.07591OpenAlexW2963887505MaRDI QIDQ1677971

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Publication date: 14 November 2017

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Abstract: In this paper, we discuss the maximum principle for a time-fractional diffusion equation ${\displaystyle }$ partial_t^alpha u(x,t) = sum_{i,j=1}^n partial_i(a_{ij}(x)partial_j u(x,t)) + c(x)u(x,t) + F(x,t), t>0, x in Omega subset {mathbb R}^n${\displaystyle }$ with the Caputo time-derivative of the order $alphain(0,1)$ 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 $c=c(x)$.

Full work available at URL: https://arxiv.org/abs/1702.07591

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