We analyze the behavior of the asymptotic dynamics of dissipative reaction-diffusion equations with Neumann boundary conditions when the domain where the equation is posed undergoes certain perturbation. We will focus on the behavior of the stationary solutions, their local unstable manifolds and the attractors.
We will consider “regular” perturbations of the domain, that is, perturbations for which the spectra of the Laplace operator behaves continuously. In this case, it turns out that if all the equilibria of the unperturbed system are nondegenerate (hyperbolic), then both the equilibria and their local unstable manifolds behave continuously under the perturbation. Exploiting the gradient properties of the flow we will show that the “attractors” also behave continuously.
We may also consider some “non-regular” perturbations of the domain. In this situation, the problem needs to be studied and the technique adapted for each particular case. An interesting example of non regular perturbations is the “dumbbell domain” which consists in two domains joined by a very thin channel which degenerates to a line segment. We will describe the results obtained for this perturbation.