Nonclassical equations of mathematical physics. Phase space of semilinear Sobolev type equations

N. A. Manakova   ; G. A. Sviridyuk

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 8 (2016) no. 3, pp. 31-51

Résumé

The article surveys the results concerning the morphology of phase spaces for semilinear Sobolev type equations. The first three paragraphs present specific boundary value problems for Sobolev type partial differential equations whose phase spaces are simple smooth Banach manifolds. The last section contains the mathematical models whose phase spaces lie on a smooth Banach manifolds with singularities. The purpose of this article is the formation of a basis for future studies of the morphology of phase spaces for semilinear Sobolev type equations. In addition, the article provides an explanation of the phenomenon of nonexistence of solutions to the Cauchy problem and the phenomenon of nonuniqueness of solutions to the Showalter–Sidorov problem for the semilinear Sobolev type equations.

Détail

Mots-clés : Sobolev type equations
Keywords: phase space, the morphology of the phase space, Banach manifold, quasistationary trajectory, Showalter–Sidorov problem, Cauchy problem, $k$-assembly Whitney.

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