The Neumann problem in an irregular domain (Q629816)

The paper studies the stability of patterns (solutions that have an interface, that is a transition layer between two constants) for the equation \(u_{t}-\Delta u=g(u) \) with Neumann boundary conditions in an irregular domain. The problem is considered as a perturbation of a limit problem posed on a set \(\Omega\) which is disconnected, but such that \(\overline{\Omega}\) is connected. The method used here is based on the degree and operator perturbation theories. One proves that in 2D dumbbell domains, stable patterns exist. Finally, one studies the convergence of the evolution equation. One estimates the difference of the semigroups in a norm containing an exponential weight with respect to time. Theoretical results are illustrated by numerical simulations of evolving and persisting interfaces.