Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients

• Published in: Applied mathematics and computation 2015-04, Vol.257, p.381-397
• Main Authors: Li, Zhiyuan, Liu, Yikan, Yamamoto, Masahiro
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Boundary value problems; Constants; Derivatives; Diffusion; Initial value problems; Initial-boundary value problem; Laplace transform; Long-time asymptotic behavior; Mathematical analysis; Mathematical models; Multinomial Mittag–Leffler function; Time-fractional diffusion equation; Well-posedness
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2014.11.073

In this paper, we investigate the well-posedness and the long-time asymptotic behavior for initial-boundary value problems for multi-term time-fractional diffusion equations. The governing equation under consideration includes a linear combination of Caputo derivatives in time with decreasing orders in (0,1) and positive constant coefficients. By exploiting several important properties of multinomial Mittag–Leffler functions, various estimates follow from the explicit solutions in form of these special functions. Then we prove the uniqueness and continuous dependency on initial values and source terms, from which we further verify the Lipschitz continuous dependency of solutions with respect to coefficients and orders of fractional derivatives. Finally, by a Laplace transform argument, it turns out that the decay rate of the solution as t→∞ is given by the minimum order of the time-fractional derivatives.