 197 Hongyu Cheng, Rafael de la Llave
 Time dependent center manifold in PDEs
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Jan 23, 19

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Abstract. We consider externally forced evolution equations. Mathematically,
these are skew systems driven by a finite dimensional evolution. Two very
common cases included in our treatment
are quasiperiodic forcing and forcing by a
stochastic process. We allow that the evolution is
a PDE and even that it is not wellposed and that it does
not define an
evolution (flow).
We first establish a general abstract theorem which, under suitable
(spectral, nondegeneracy, smoothness, etc) assumptions, establishes the
existence of a ``timedependent invariant manifold'' (TDIM). These
manifolds evolve with the forcing. They
are such that the original equation is always tangent to a vector field in
the manifold. Hence, for initial data in the TDIM, the original
equation is equivalent to an ordinary differential equation. This
allows us to define families of solutions of the full equation by
studying the solutions of a finite dimensional system. Note that this
strategy may
apply even if the original equation is ill posed and does not admit solutions
for arbitrary initial conditions (the TDIM selects initial conditions for
which solutions exist). It also allows that the TDIM is infinite dimensional.
Secondly, we construct the center manifold for skew systems driven by the external forcing.
Thirdly, we present concrete applications of the abstract result
to the differential equations whose linear operators are exponential trichotomy subject to quasiperiodic
perturbations. The use of TDIM allows us to establish the existence of
quasiperiodic solutions and to study the effect of resonances.
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