Convergence theorems for equations related to phase transitions

We study long time behavior of solutions to the quasi-steady approximation of the equation of viscoelasticity with capillarity. We consider the equation with a number of boundary conditions including such that the set of equilibria is necessarily at least one-dimensional. We prove that any solution converges to an equilibrium point. We admit some kind of initial data with infinite energy. We use tools based on the assumption that the nonlinear term is real analytic and our observation that the equation in question generates a gradient flow in $H^{-1}$.

We also remark that the same method applies (with changes) to Cahn-Hilliard equation.