Slow manifolds for dissipative dynamical systems
The author considers the initial value problem \[ \dot u=- Au+ {(Au,u)\over (u,u)},\quad u(0)= u_0\tag{\(*\)} \] in a Hilbert space \(H\) with inner product \((\cdot,\cdot)\), where \(A= A^*\) is a positive self-adjoint operator in \(H\) with spectrum \(\sigma(A)\). He derives conditions on \(\sigma(A)\) such that \((*)\) has a unique global solution \(\widetilde u(t)\) satisfying \(\lim_{t\to\infty}\widetilde u(t)= v\) in the strong sense. If \(v\) is an eigenvector of \(A\) to the smallest eigenvalue of \(A\), then the corresponding eigenspace is called the ``slow'' invariant manifold of \(\dot u=- Au\).
scientific article; zbMATH DE number 5651228