Consider the Dirichlet problem for the reaction-diffusion equation
Here is a bounded domain in with smooth boundary and is a sufficiently regular function on . Problem (1) defines a local semiflow on an appropriate Banach space, for example, the Sobolev space with . This semiflow is gradient-like: the energy functional
where is the antiderivative of with respect to , decreases along nonconstant trajectories. In higher space dimensions, stable and unstable manifolds of hyperbolic equilibria can intersect nontransversally. One of the main objectives of the present paper is to prove that generically this cannot happen. To formulate the result precisely, let be a positive integer and let denote the space of all functions endowed with the Whitney topology. This is the topology in which the collection of all the sets
where is a positive continuous function on , forms a neighborhood basis of an element . Recall that is a Baire space: any residual set is dens in . Our main result reads as follows.
Theorem. There is a residual set in such that for any all equilibria of (1) are hyperbolic and if , are any two such equilibria then their stable and unstable manifolds intersect transversally.