# Finite-dimensional limiting dynamics for dissipative parabolic equations

## Abstract

For a broad class of semilinear parabolic equations with compact attractor A in a Banach space E the problem of a description of the limiting phase dynamics (the dynamics on A) of a corresponding system of ordinary differential equations in R{sup N} is solved in purely topological terms. It is established that the limiting dynamics for a parabolic equation is finite-dimensional if and only if its attractor can be embedded in a sufficiently smooth finite-dimensional submanifold M subset of E. Some other criteria are obtained for the finite dimensionality of the limiting dynamics: a) the vector field of the equation satisfies a Lipschitz condition on A; b) the phase semiflow extends on A to a Lipschitz flow; c) the attractor A has a finite-dimensional Lipschitz Cartesian structure. It is also shown that the vector field of a semilinear parabolic equation is always Holder on the attractor.

- Authors:

- All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21202922

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 191; Journal Issue: 3; Other Information: DOI: 10.1070/SM2000v191n03ABEH000466; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATTRACTORS; BANACH SPACE; DIFFERENTIAL EQUATIONS; TOPOLOGY; VECTOR FIELDS

### Citation Formats

```
Romanov, A V.
```*Finite-dimensional limiting dynamics for dissipative parabolic equations*. United States: N. p., 2000.
Web. doi:10.1070/SM2000V191N03ABEH000466; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Romanov, A V.
```*Finite-dimensional limiting dynamics for dissipative parabolic equations*. United States. doi:10.1070/SM2000V191N03ABEH000466; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Romanov, A V. Sun .
"Finite-dimensional limiting dynamics for dissipative parabolic equations". United States. doi:10.1070/SM2000V191N03ABEH000466; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21202922,
```

title = {Finite-dimensional limiting dynamics for dissipative parabolic equations},

author = {Romanov, A V},

abstractNote = {For a broad class of semilinear parabolic equations with compact attractor A in a Banach space E the problem of a description of the limiting phase dynamics (the dynamics on A) of a corresponding system of ordinary differential equations in R{sup N} is solved in purely topological terms. It is established that the limiting dynamics for a parabolic equation is finite-dimensional if and only if its attractor can be embedded in a sufficiently smooth finite-dimensional submanifold M subset of E. Some other criteria are obtained for the finite dimensionality of the limiting dynamics: a) the vector field of the equation satisfies a Lipschitz condition on A; b) the phase semiflow extends on A to a Lipschitz flow; c) the attractor A has a finite-dimensional Lipschitz Cartesian structure. It is also shown that the vector field of a semilinear parabolic equation is always Holder on the attractor.},

doi = {10.1070/SM2000V191N03ABEH000466; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 3,

volume = 191,

place = {United States},

year = {2000},

month = {4}

}