Maximum principle for time-fractional parabolic equations with a reaction coefficient of arbitrary sign

TL;DR

This paper establishes a weak maximum principle for time-fractional parabolic equations with arbitrary reaction coefficient signs, using an elementary proof that applies to various boundary conditions and coefficients.

Contribution

Provides a novel, elementary proof of the weak maximum principle for time-fractional parabolic equations without sign restrictions on the reaction coefficient.

Findings

Maximum principle holds without sign restrictions on reaction coefficient

Proof extends to weak solutions and diverse boundary conditions

Applicable to variable-coefficient and variable-order fractional equations

Abstract

We consider time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction coefficient. This proof is also extended for weak solutions, as well as for various types of boundary conditions and variable-coefficient variable-order multiterm time-fractional parabolic equations.