Amorphous molecular beam epitaxy: global solutions and absorbing sets

The parabolic equation ut + uxxxx + uxx = -(|ux|α)xx, α > 1, is studied under the boundary conditions ux|∂Ω = uxxx| ∂Ω = 0 in a bounded real interval Ω. Solutions from two different regularity classes are considered: It is shown that unique mild solutions exist locally in time for any α > 1 and initial data u 0 ∈ W1,q(Ω) (q > α), and that they are global if α ≤ 5/3. Furthermore, from a semidiscrete approximation scheme global weak solutions are constructed for α < 10/3, and for suitable transforms of such solutions the existence of a bounded absorbing set in L1(Ω) is proved for α ∈ [2, 10/3). The article closes with some numerical examples which do not only document the roughening and coarsening phenomena expected for thin film growth, but also illustrate our results about absorbing sets. © 2005 Cambridge University Press.