Properties of Solutions to the Full Fractional Heat Operator Equation

TL;DR

This paper investigates the properties of solutions to a fully fractional heat equation, proving nonexistence of positive bounded solutions under certain conditions and introducing mathematical tools applicable to fractional operators.

Contribution

It provides new nonexistence results for positive solutions and develops mathematical tools for analyzing fractional heat equations with nonlinearities.

Findings

No positive bounded solutions under certain nonlinearity conditions.

Solution monotonicity along the first direction.

Mathematical tools for fractional operator analysis.

Abstract

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator . Under certain assumptions of the indefinite nonlinearity and its weight, we prove that there is no positive bounded solution, which is based on the monotonicity of the solution along the first direction that is proved by employing the method of moving planes. Besides, if the weight satisfy other conditions, we come to different conclusions according to the behavior of the nonlinearity at infinity. To overcome the difficulties caused by the operator, we lead in some mathematics tools that, as we believe, will be useful in studying problems involving other fractional operators or nonlinearities.