On the maximum principle for a time-fractional diffusion equation

Abstract

In this paper, we discuss the maximum principle for a time-fractional diffusion equation [Equation presented here], with the Caputo time-derivative of the order α ∈ (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).