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

Yuncheng You

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.