  In this paper, we discuss initial-boundary value problems for linear
diffusion equation with multiple time-fractional derivatives. By means of the
Mittag-Leffler function and the eigenfunction expansion, we reduce the problem
to an integral equation for a solution, and we apply the fixed-point theorem to
prove the unique existence and the H\"older regularity of solution. For the
case of the homogeneous equation, the solution can be analytically extended to
a sector $\{z\in\mathbb{C};z\neq0,|\arg z|<\frac{\pi}{2}\}$. In the case where
all the coefficients of the time-fractional derivatives are positive constants,
by the Laplace transform and the analyticity, we can prove that if a function
satisfies the fractional diffusion equation and the homogeneous Neumann
boundary condition on arbitrary subboundary as well as the homogeneous
Dirichlet boundary condition on the whole boundary, then it vanishes
identically.
