INERTIAL MANIFOLDS FOR NONAUTONOMOUS SKEW PRODUCT SEMIFLOWS

Yuncheng You (U. S. A.)

Received September 20, 2006

Abstract

The mild solution of a nonlinear nonautonomous evolutionary equation can be formulated as a skew product semiflow in a product phase space. Under a spectral gap condition, it is shown that there exists an inertial manifold for this skew product semiflow. Instead of the Lyapunov-Perron method, the proof is fulfilled via the approach of conic invariance and incrementally exponential dichotomy and based on two conic differential inequalities. The construction of inertial manifold is made through an exponentially tracking integral manifold, in which the pullback is achieved also by the incremental dichotomy and a homotopy lemma. An illustration of the applications is shown by nonautonomous reaction-diffusion equations.

Keywords and phrases: global dynamics, inertial manifold, integral manifold, nonautonomous evolutionary equation, skew product semiflow.