  A doubly nonlinear parabolic equation of the form $\alpha(u_t)-\Delta
u+W'(u)= f$, complemented with initial and either Dirichlet or Neumann
homogeneous boundary conditions, is addressed. The two nonlinearities are given
by the maximal monotone function $\alpha$ and by the derivative $W'$ of a
smooth but possibly nonconvex potential $W$; $f$ is a known source. After
defining a proper notion of solution and recalling a related existence result,
we show that from any initial datum emanates at least one solution which gains
further regularity for $t>0$. Such "regularizing solutions" constitute a
semiflow $S$ for which uniqueness is satisfied for strictly positive times and
we can study long time behavior properties. In particular, we can prove
existence of both global and exponential attractors and investigate the
structure of $\omega$-limits of single trajectories.
