Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. (English) Zbl 0692.35053

Summary: We study the long-time behavior of a class of nonlinear dissipative partial differential equations. By means of the Lyapunov-Perron method, we show that the equation has an inertial manifold, provided that certain gap condition in the spectrum of the linear part of the equation is satisfied. We verify that the constructed inertial manifold has the property of exponential tracking (i.e., stability with asymptotic phase, or asymptotic completeness), which makes it a faithful representative to the relevant long-time dynamics of the equation. The second feature of this paper is the introduction of a modified Galerkin approximation for analyzing the original PDE. In an illustrative example (which we believe to be typical), we show that this modified Galerkin approximation yields a smaller error than the standard Galerkin approximation.


35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
37-XX Dynamical systems and ergodic theory
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