$ u_t  = Lu + f(t,x,u,\nabla u)$,     $ x \in \Omega $,   $ t>0 $,

under suitable boundary conditions. Here $L$ is a second-order
elliptic operator and $\Omega$ is a bounded domain. The topics
included  in the  note reflect the ones  discussed at the  author's
lectures at the workshop ``New Trends in Nonlinear Partial
Differential Equations'', held at the Ryukoku University in 1999,
specifically:

    Typical behavior of solutions

    Possible dynamics on invariant manifolds

    Equations on symmetric domains