2023/06/20 by Lingwei Ma, Ma, Lingwei, Yahong Guo +3 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2306.11554
openalex publication_date 2023/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation (∂t-Δ)s u(x,t) =f(u(x, t)), (x, t)∈ B1(0)× ℝ, subject to the vanishing exterior condition, are radially symmetric and strictly decreasing with respect to the origin in B1(0) for any t∈ ℝ. Another one is to establish the Liouville theorem for homogeneous master equation (∂t-Δ)s u(x,t)=0 , in ℝn×ℝ, which states that all bounded solutions must be constant. We propose a new methodology for a direct method of moving planes applicable to the fully fractional heat operator (∂t-Δ)s, and the proof of our main results based on this direct method involves the perturbation technique, limit argument as well as Fourier transform. This study opens up a way to investigate the geometric behavior of master equations, and provides valuable insights for establishing qualitative properties of solutions and even for deriving important Liouville theorems for other types of fractional order parabolic equations.