Three-dimensional stability of stationary reaction balls is investigated in a system
consisting of an autocatalysis accompanied by a slow decay of the autocatalyst. Radially
stable stationary spherical structures become unstable to three-dimensional perturbations
at small decay rate when the radius of the reaction ball is sufficiently large and
the reactant diffuses faster than the autocatalyst. A thorough linear stability analysis
and simulations in three spatial dimensions are carried out in the simplest system
sustaining stationary reaction balls.