Cauchy Problem with Subcritical Nonlinearity

The semilinear parabolic system onRⁿwith subcritical nonlinearity is studied as an abstract evolutionary equation in a Banach spaceX ? ^p(Rⁿ). As in the case of bounded domains the existence of a strongly continuous semigroup of globalX^α-solutions and its dissipativeness is shown to follow from a single estimate of solutions. Despite the lack of compactness of the Sobolev embeddings inRⁿthe compactness of trajectories of the semigroup is proved using the auxiliary estimate of solutions in a weighted space ^p(Rⁿ, (1 + |x|²)^ν), and the existence of a global attractor is also shown. Thus this paper generalizes earlier considerations and provides a basis for further applications.