Inertial manifolds and the cone condition. (English) Zbl 0787.34036

Summary: The “cone condition”, used in passing in many proofs of the existence of inertial manifolds, is examined in more detail. Invariant manifolds for dissipative flows can be obtained directly using no other dynamical information. After finding a condition for the exponential attraction of trajectories to such a manifold, a cone invariance property is used to show the existence of orbits on the manifold which track a given orbit of the flow. This leads to a concise proof which guarantees the existence of inertial manifolds with the asymptotic completeness property. Furthermore it is shown that the “strong squeezing property” implies directly the existence of such an inertial manifold. There follows a brief discussion of the rôle of the cone condition in the Lyapunov-Perron fixed point method of proof, and a comparison with previous results.


34C30 Manifolds of solutions of ODE (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations