Properties of Solutions to the Full Fractional Heat Operator Equation

2025/11/10 by Haipeng, Lu, Mei, Yu
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.07153

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.

Citations

• Master equations with indefinite nonlinearities
• Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games
• Fractional Orlicz-Sobolev embeddings
• Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation
• The influence of a line with fast diffusion on Fisher-KPP propagation
• Fractional Sobolev Inequalities: Symmetrization, Isoperimetry and Interpolation
• Extension problem and Harnack's inequality for some fractional operators
• Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation
• Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

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• Optimal Existence and Uniqueness Theory for the Fractional Heat Equation
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