On the existence of patterns in reaction-diffusion problems with Dirichlet boundary conditions

Consider a general reaction-diffusion problem, $u_t = \Delta u + f(x, u, u_x)$, on a revolution surface or in an $n$-dimensional ball with Dirichlet boundary conditions. In this work, we provide conditions related to the geometry of the domain and the spatial heterogeneities of the problem that ensure the existence or not of a non-constant stationary stable solution. Several applications are presented, particularly with regard to the Allen–Cahn, Fisher–KPP and sine-Gordon equations.