Finite-dimensional limiting dynamics of dissipative parabolic equations. (English. Russian original) Zbl 0963.37074

Romanov, A. V.

Sb. Math. 191, No. 3, 415-429 (2000); translation from Mat. Sb. 191, No. 3, 99-112 (2000).

This paper is devoted to the finite-dimensional behaviour of solutions to parabolic equations for large time. The author is mainly interested in the behaviour of evolution equations that is intermediate between finite dimensionality of the attractor and existence of an inertial manifold. The main result of the author deals with the possibility of embedding the attractor \(A\) in a sufficiently smooth finite-dimensional submanifold \({\mathcal M}\subset E^\alpha\), guaranteeing that the limiting dynamics for a broad class of semilinear parabolic equations (1) \(\partial_t u=-Au+Fu\) is finite-dimensional. Here \(A\) is a linear sectorial operator acting in a Banach space \(E\), \(E^\alpha= D(A^\alpha)\) and \(F\) is a nonlinear term.

Reviewer: Messoud Efendiev (Berlin)

MSC:

37L30	Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
35B41	Attractors
35K57	Reaction-diffusion equations
35B42	Inertial manifolds

Keywords:

global attractor; inertial manifold; semilinear parabolic equations; finite dimension of attractor