### Peter Polácik

## Asymptotic behavior and dynamics on invariant manifolds

The survey will mainly be concerned with scalar equations, although systems
will also be mentioned occasionally. In a general form, the equations we will
consider read as follows:

**u**_{t} = F(x, t, u, Du, D^{2}u),
**x** in **Omega**,
where **Omega** is a smooth domain and
**F** is a real-valued function satisfying appropriate regularity and
parabolicity assumptions. We shall mostly restrict our attention
to bounded domains and periodically time-dependent (or autonomous)
nonlinearities. The survey will provide an account of results and
techniques in the study of the large-time behavior of bounded solutions.

Among the very few general tools that are available for the study of
asymptotics, invariant manifolds, maximum principles and Lyapunov functionals
are probably the most frequently employed ones. We start the survey by
reviewing properties of solutions that one can obtain invoking the maximum
principle or a gradient-like structure. Usually there are additional structural
assumptions to help. We intend to discuss the following:

- Asymptotics in one space dimension (convergence and
Poincaré-Bendixson theorems, Floquet bundles and perturbations).
- Asymptotics of positive solutions on higher dimensional symmetric
domains (asymptotic symmetrization, spatio-temporal asymptotics).
- Equations with a gradient structure (convergence theorems
via analyticity or normal hyperbolicity).
- General equations and monotone dynamical systems (asymptotic
periodicity of a typical trajectory).

In these results, the asymptotics of the solutions is always a relatively
simple one (convergence to an equilibrium or periodic orbit). Although this
is what one typically observes for this type of equations, such a simple
behavior is not guaranteed in general. Center manifolds and, more
specifically, the technique of realization of vector fields on center
manifolds, appear to be very useful for understanding of what kind of
dynamics can be encountered. We shall discuss these methods and review
some of the results achieved. In particular, we mention

- existence of chaotic dynamics; existence of trajectories with high
dimensional limit sets;
- heat equations that have a gradient structure, but do not
have all bounded trajectories convergent.

*
Peter Polácik *

Jan 15 1998