The roles of diffusivity and curvature in patterns on surfaces of revolution

We address the question of finding sufficient conditions for existence as well as nonexistence of nonconstant stable stationary solution to the diffusion equation ut=div(a∇u)+f(u)ut=div(a∇u)+f(u) on a surface of revolution with and without boundary. Conditions found relate the diffusivity function a and the geometry of the surface where diffusion takes place. In the case where f is a bistable function, necessary conditions for the development of inner transition layers are given.