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We prove that any global bounded solution of a phase field model with memory terms tends to a single equilibrium state for large times. Because of the memory effects, the energy is not a Lyapunov function for the problem and the set of equilibria may contain a nontrivial continuum of stationary states. The method we develop is applicable to a more general class of equations containing memory terms.

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Communicated by P. Colli SUMMARY We prove that any global bounded solution of a phase ÿeld model tends to a single equilibrium state for large times though the set of equilibria may contain a nontrivial continuum of stationary states. The problem has a partial variational structure, speciÿcally, only the elliptic part of the ÿrst equation represents an Euler-Lagrange equation while the second does not. This requires some modiÿcations in comparison with standard methods used to attack this kind of problems.

We consider a nonlinear Neumann problem driven by the p-Laplacian differential operator and having a p-superlinear nonlinearity. Using truncation techniques combined with the method of upper-lower solutions and variational arguments based on critical point theory, we prove the existence of five nontrivial smooth solutions, two positive, two negative and one nodal. For the semilinear (i.e., p = 2) problem, using critical groups we produce a second nodal solution.