Patterns on surfaces of revolution in a diffusion problem with variable diffusivity

In this article we study the existence of non-constant stable stationary solutions to the the diffusion equation $u_t=\hbox{div}(a \nabla u)+f(u)$ on a surface of revolution whose border is supplied with zero Neumann boundary condition. Sufficient conditions on the geometry of the surface and on the diffusivity function $a$ are given for the existence of a function f such the problem possesses such solutions