The asymptotic radial symmetry and monotonicity of the solution for logarithmic laplacian parabolic equation

In this paper, we investigate the positive solutions to the following Logarithmic Laplacian parabolic equation$ {align} {cases}∂ u/∂ t(x,t)+(-)^Lu(x,t) = f(t,u(x,t)),&(x,t)∈ B_1(0)×(0,∞),\(x,t) = 0,&(x,t)∈ B_1^c(0)×(0,∞).{cases} {align} $Here, $ B_1(0)$ reprents the unit sphere in $ R^N $. First, we establish a series of key maximum principles, including asymptotic narrow region principle, asymptotic maximum principle and asymptotic strong maximum principle for the parabolic equation with (-Δ)L. On this basis, we propose an asymptotic method of moving planes appropriate for the parabolic equation with (-Δ)L, and obtain the asymptotic radial symmetry and monotonicity of the positive solutions to this equation. Finally, we prove a Hopf's Lemma for (-Δ)L and an asymptotic Hopf's Lemma applicable to antisymmetric functions.