Maximum Principle and Its Application for the Time-Fractional Diffusion Equations
Fractional Calculus and Applied Analysis, Tome 14 (2011) no. 1, p. 110.

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MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversaryIn the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional derivative is shown to possess a suitable generalization of the extremum principle well-known for ordinary derivative. As an application, the maximum principle is used to get some a priori estimates for solutions of initial-boundary-value problems for the generalized time-fractional diffusion equations and then to prove uniqueness of their solutions.
Classification : 45K05, 35R11, 35B45, 35B50, 35K99, 35-02
Mots-clés : Time-Fractional Diffusion Equation, Time-Fractional Multiterm Diffusion Equation, Time-Fractional Diffusion Equation of Distributed Order, Extremum Principle, Caputo Fractional Derivative, Generalized Riemann-Liouville Fractional Derivative, Initial-Boundary-Value Problems, Maximum Principle, Uniqueness Results, time-fractional diffusion equation, time-fractional multi-term diffusion equation, time-fractional diffusion equation of distributed order, extremum principle, Caputo fractional derivative generalized Riemann-Liouville fractional derivative initial-boundary-value problems maximum principle uniqueness results 
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Luchko, Yury. Maximum Principle and Its Application for the Time-Fractional Diffusion Equations. Fractional Calculus and Applied Analysis, Tome 14 (2011) no. 1, p. 110. https://geodesic-test.mathdoc.fr/item/FCAA_2011__14_1_219664/