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:

ut = F(x, t, u, Du, D2u),    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:

  1. Asymptotics in one space dimension (convergence and Poincaré-Bendixson theorems, Floquet bundles and perturbations).
  2. Asymptotics of positive solutions on higher dimensional symmetric domains (asymptotic symmetrization, spatio-temporal asymptotics).
  3. Equations with a gradient structure (convergence theorems via analyticity or normal hyperbolicity).
  4. 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

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

Peter Polácik

Jan 15 1998