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Invariant manifolds of semilinear Sobolev type equations. (English) Zbl 1492.35003

Summary: The article is devoted to a review of the author’s results in studying the stability of semilinear Sobolev type equations with a relatively bounded operator. We consider the initial-boundary value problems for the Hoff equation, for the Oskolkov equation of nonlinear fluid filtration, for the Oskolkov equation of plane-parallel fluid flow, for the Benjamin-Bon-Mahoney equation. Under an appropriate choice of function spaces, these problems can be considered as special cases of the Cauchy problem for a semilinear Sobolev type equation. When studying stability, we use phase space methods based on the theory of degenerate (semi)groups of operators and apply a generalization of the classical Hadamard-Perron theorem. We show the existence of stable and unstable invariant manifolds modeled by stable and unstable invariant spaces of the linear part of the Sobolev type equations in the case when the phase space is simple and the relative spectrum and the imaginary axis do not have common points.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B42 Inertial manifolds
35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35S10 Initial value problems for PDEs with pseudodifferential operators
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
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