Continuous and Distributed Systems by Mikhail Z. Zgurovsky & Victor A. Sadovnichiy

Author:Mikhail Z. Zgurovsky & Victor A. Sadovnichiy
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof

The statement of lemma follows from [12] and Theorem 11.1.

Lemma 11.8

Kasyanov [12] Let be a global attractor from Theorem 11.2. Then

(11.29)

Theorem 11.3

Let be a global attractor from Theorem 11.2. Then there exists the trajectory attractor in the space . At that the next formula holds:

(11.30)

Proof

The statement of theorem follows from Theorem 11.1 and [12].

11.4 Application

Consider an example of the class of nonlinear boundary value problems for which we can investigate the dynamics of solutions as . Note that in discussion we do not claim generality.

Let , , , , , be a bounded domain with rather smooth boundary . We denote a number of differentiations by of order (correspondingly of order ) by (correspondingly by ). Let be a family of real functions (), defined in and satisfying the next properties:

( ) for a.e. the function is continuous one in ;

( ) the function is measurable one in ;

( ) exist such and , that for a.e. ,

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