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Stability of some solutions for elliptic equations on a cylindrical domain

We study analytically a class of solutions for the elliptic equation where α>0 and ϵ is a small parameter. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every α>0, it contains solutions that are defined for large values of time and they are very close (of order O(ϵ)) to a linear torus for long times (of order O(ϵ−1)). The proof uses the fact that the equation leaves invariant a smooth centre manifold and, for the restriction of the system to the centre manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.