On the maximum principle for a time-fractional diffusion equation

In this paper, we discuss the maximum principle for a time-fractional diffusion equation $${\mathrm{\partial }}_t^{\alpha }u(x,t)=\sum _{i,j=1}^n{\mathrm{\partial }}_i(a_{ij}(x){\mathrm{\partial }}_{j}u(x,t))+c(x)u(x,t)+F(x,t),t>0,x\in \mathrm{\Omega }\subset {\mathbb{R}}^n$$ 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 $c=c(x)$.