In this well-written paper, the existence of a global attractor for nonlinear reaction-diffusion equations in of the form
with initial data
is shown. Here, the nonlinearity is allowed to have polynomial growth of arbitrary order and in the inhomogeneous term, , , are distributional derivatives of , .
In order to obtain this for the problem , two difficulties appear: (1) The regularity of its solutions is not sufficiently high to apply appropriate embedding theorems. (2) It is hard to get continuity of the associated semigroup in the -topology without restriction on .
Thus, for abstract semigroups in the authors derive a sufficient criterion that a global attractor in also attracts bounded sets of w.r.t. the -norm.
Using a new method based on a priori estimates, this criterion applies to show that the semigroup in associated with , possesses a -global attractor in the sense that is nonempty, compact, invariant in and attracts every bounded subset of in the -norm.