Abstract
In this paper, we discuss the maximum principle for a time-fractional diffusion equation
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).
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Dedicated to Professor Virginia Kiryakova on the occasion of her 65th birthday and the 20th anniversary of FCAA
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Luchko, Y., Yamamoto, M. On the maximum principle for a time-fractional diffusion equation. FCAA 20, 1131–1145 (2017). https://doi.org/10.1515/fca-2017-0060
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DOI: https://doi.org/10.1515/fca-2017-0060