Time dependent center manifold in PDEs.

*(English)*Zbl 07273493Summary: We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution).

We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a “time-dependent 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 quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.

We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a “time-dependent 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 quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.

##### MSC:

35B42 | Inertial manifolds |

35B15 | Almost and pseudo-almost periodic solutions to PDEs |

35R25 | Ill-posed problems for PDEs |

37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |

35J60 | Nonlinear elliptic equations |

47J06 | Nonlinear ill-posed problems |

37L25 | Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems |

##### Keywords:

reduction principles; skew systems; quasi-periodic forcing; forcing by a stochastic process; exponential trichotomy
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\textit{H. Cheng} and \textit{R. de la Llave}, Discrete Contin. Dyn. Syst. 40, No. 12, 6709--6745 (2020; Zbl 07273493)

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