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

$$\partial _t^\alpha u\left( {x,t} \right) = \sum\limits_{i,j = 1}^n {{\partial _i}\left( {{a_{ij}}\left( x \right){\partial _j}u\left( {x,t} \right)} \right) + c\left( x \right)u\left( {x,t} \right)} + F\left( {x,t} \right),t >0,x \in \Omega \subset {^n},$$

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).