Asymptotic symmetry of solutions for reaction-diffusion equations via elliptic geometry

In this paper, we investigated the asymptotic symmetry and monotonicity of positive solutions to a reaction-diffusion equation in the unit ball, utilizing techniques from elliptic geometry. First, we discussed the properties of solutions in the elliptic space. Then, we established crucial principles, including the asymptotic narrow region principle. Finally, we employed the method of moving planes to demonstrate the asymptotic symmetry of the solutions.