Cauchy Problem with Subcritical Nonlinearity

The semilinear parabolic system on Rn with subcritical nonlinearity is studied as an abstract evolutionary equation in a Banach space X[L p Rn.. As in the case of bounded domains the existence of a strongly continuous semigroup of global Xa-solutions and its dissipativeness is shown to follow from a single estimate of solutions. Despite the lack of compactness of the Sobolev embeddings in Rn the compactness of trajectories of the semigroup is proved using the auxiliary estimate of solutions in a weighted space L p Rn, 1q< x < 2 .n ., and the existence of a global attractor is also shown. Thus this paper generalizes earlier considerations and provides a basis for further applications.