Amorphous molecular beam epitaxy: global solutions and absorbing sets

The parabolic equation ut + uxxxx + uxx = - (|ux|^(alpha))xx, (alpha)>1, is studied under the boundary conditions ux|(partial)(Omega)=uxxx|(partial)(Omega)=0 in a bounded real interval (Omega). Solutions from two different regularity classes are considered: It is shown that unique mild solutions exist locally in time for any (alpha)>1 and initial data u0 in W^1,q(Omega) q(alpha), and that they are global if (alpha)<=5/3. Furthermore, from a semidiscrete approximation scheme global weak solutions are constructed for (alpha) < 10/3, and for suitable transforms of such solutions the existence of a bounded absorbing set in L^1(Omega) is proved for (alpha)in [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.