Center-Unstable Manifolds for PDEs on Infinite Strips

Rouslan Krechetnikov , University of California at Santa Barbara

Samuel Paolucci, University of Notre Dame

The well-established center manifold theory for finite-dimensional systems [Ca] has been extended by a number of authors to countable infinite-dimensional systems. The latter usually arise from PDEs defined on compact domains (separable spaces). In this situation the global attracting set is compact with finite Hausdorff dimension. One should recognize that the analogous theory for PDEs on extended domains is far from being at the same level of elaboration. Contrary to the compact case, with increase of the spatial extension of the system, the dimension of the attractor grows and becomes infinite in the limit of infinite extension. The main obstacle for application of standard bifurcation theory is the presence of dense discrete or continuous spectra, which prevent from having spectral gap usually required for the application of center manifold theory. The first theoretical result on the reduction of such problems is due to Kirchgassner [Ki], who applies the standard center manifold theory to stationary (non-evolutionary) problems of elliptic type by taking the unbounded spatial direction as an evolutionary-like variable. This idea has been extended by Mielke [Mi], who uses the same approach applied to the construction of essential solutions of unsteady problems by considering both time and space variables on unbounded domains (note: not semi-bounded). This essential set of solutions, therefore, must exist for negative time, thus excluding treatment of initial value problems and the stability of these essential solutions.

In this contribution we discuss the construction of center-unstable manifolds for PDEs defined on infinite strips: domains with at least one bounded and one unbounded dimensions. This class of problems allows a natural extension of the center-unstable manifold theory on the basis of the well-known center-stable theorem for infinite-dimensional systems [Ga]. The new approach represents progress in the reduction of such PDEs over existing methods [Mi]. In particular, our theory allows the treatment of initial value problems and, consequently, the investigation of the stability of the absorbing set. It yields as output a reduced problem which is local in time, in contrast to a non-local one given by Mielke [Mi]. As a consequence, our method gives results that are no weaker than those given by the classic center manifold theory (when there is no unbounded dimension). We also discuss certain peculiar properties of the reduction which are analogous to those arising in center manifold theory for ODEs (e.g., resonance conditions).

[Ca] J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, vol. 35, Springer-Verlag, Berlin/New York (1981).

[Ga] Th. Gallay, A center-stable manifold theorem for differential equations in Banach spaces, Commun. Math. Phys. 152, 249--268 (1993).

[Ki] K. Kirchgassner, Wave solutions of reversible systems and applications, J. Diff. Eq. 45, 113--127 (1982).

[Mi] A. Mielke, Reduction of PDEs on domains with several unbounded directions: A first step towards modulation equations, Z. Angew. Math. Phys. 43, 449--470 (1992).