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For $0 < \alpha < 1$ and bounded domains in $\mathbf{R}^N$ the hard analysis for divergence form time-fractional diffusion equations
${}^{RL}D^\alpha_t u - div( A(t,x)Du) = 0$
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:  ${}^{RL}D^{\alpha}_t u = {}^CD^{\alpha}_t u - g(t) u(0)$.  For $1<\alpha<2$ 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.