Lie Symmetry Analysis and Invariant Solutions of the Diffusion Equation on the Sphere with Linear Reaction

This paper investigates the linear reaction–diffusion equation on the unit sphere by means of Lie point symmetry analysis. The determining equations show that the finite–dimensional Lie point symmetry algebra is five-dimensional and is generated by the three rotational Killing fields, time translation, and scaling, together with the infinite–dimensional superposition ideal. An optimal system of one–dimensional subalgebras is constructed and used, together with a two–stage reduction procedure based on commuting generators, to derive inequivalent similarity reductions and explicit invariant solution families. The stationary reductions lead to Legendre-type ordinary differential equations, and the global smoothness requirement on the sphere selects the polynomial branch corresponding to spherical harmonics and the associated eigenvalue quantization. In addition, mixed space– time reductions produce explicit non–stationary invariant patterns, including a locally defined family generated by a combined rotation–time–scaling symmetry. The results provide a symmetry-based framework that complements the classical spectral description of diffusion on the sphere.