Asymptotic Compactness and Attractors for Phase-Field Equations in R3

In this paper we study the asymptotic behaviour of solutions of the phase-field system on an unbounded domain. We do not assume conditions on the nonlinear term ensuring the uniqueness of the Cauchy problem, so that we have to work with multivalued semiflows rather than with semigroups of operators. In this way we prove the existence of a global attractor by considering the convergence in an appropriate weighted space. This result is also new for more restrictive conditions, which guarantee the uniqueness of solutions.