On the maximum principle for a time-fractional diffusion equation

TL;DR

This paper establishes a maximum principle for weak solutions of a time-fractional diffusion equation with Caputo derivative, extending previous results to fractional Sobolev spaces and proving non-negativity and monotonicity under broader conditions.

Contribution

It introduces a maximum principle for weak solutions in fractional Sobolev spaces and demonstrates non-negativity and monotonicity without sign restrictions on certain coefficients.

Findings

Maximum principle for weak solutions in fractional Sobolev spaces.

Non-negativity of solutions for non-negative source functions.

Monotonicity of solutions with respect to the coefficient c(x).

Abstract

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 $c=c(x)$.