For and bounded domains in the hard analysis for divergence form time-fractional diffusion equations
along with a weak Harnack inequality and applications to a maximum principle and a Liouville-type theorem is proven by Rico Zacher (ZacherFracHarnack2010). He follows the Moser iteration technique and uses Yosida approximation scheme for the Riemann-Liouville fractional derivative. His methods yield the same results for the Caputo derivative which differs from the RL derivative by a fixed function of time only: . For existence and uniqueness for the fractional wave equation are addressed using entirely different techniques by Kian and Yamamoto (Kian-Yamamoto-SemilinearFracWave2015) The latter authors work in the Hilbert space spanned the eigenfunctions of the second order operator and prove a Strichartz estimate directly using estimates of the Mittag-Leffler function.