  In this paper, we discuss the maximum principle for a time-fractional
diffusion equation $$ \partial_t^\alpha u(x,t) = \sum_{i,j=1}^n
\partial_i(a_{ij}(x)\partial_j u(x,t)) + c(x)u(x,t) + F(x,t),\ t>0,\ x \in
\Omega \subset {\mathbb R}^n$$ with the Caputo time-derivative of the order
$\alpha \in (0,1)$ in the case of the homogeneous Dirichlet boundary condition.
Compared to the already published results, our findings have two important
special features. First, we derive a maximum principle for a suitably defined
weak solution in the fractional Sobolev spaces, not for the strong solution.
Second, for the non-negative source functions $F = F(x,t)$ we prove the
non-negativity of the weak solution to the problem under consideration without
any restrictions on the sign of the coefficient $c=c(x)$ by the derivative of
order zero in the spatial differential operator. Moreover, we prove the
monotonicity of the solution with respect to the coefficient $c=c(x)$.
