International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 1, Pages 1-24
Approximate inertial manifolds for nonlinear parabolic equations via steady-state determining mapping
Department of mathematics, University of South Florida, Tampa 33620, FL, USA
Received 24 August 1992; Revised 1 December 1993
Copyright © 1995 Yuncheng You. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For nonlinear parabolic evolution equations, it is proved that, under the assumptions of local
Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial
manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the
graph of the steady-state determining mapping based on the spectral decomposition. It is also shown that
the thickness of the exponentially attracting neighborhood of the AIM converges to zero at a fractional
power rate as the dimension of the AIM increases. Applications of the obtained results to Burgers'
equation, higher dimensional reaction-diffusion equations, 2D Ginzburg-Landau equations, and axially
symmetric Kuramoto-Sivashinsky equations in annular domains are included.