Investigation of the uniqueness solution of the Showalter–Sidorov problem for the mathematical Hoff model. Phase space morphology

N. G. Nikolaeva   ; O. V. Gavrilova   ; N. A. Manakova

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 17 (2024) no. 1, pp. 49-63

Résumé

The study of the phase space morphology of the mathematical model deformation of an I-beam, which lies on smooth Banach manifolds with singularities ($k$-Whitney assembly) depending on the parameters of the problem, is devoted to the paper. The mathematical model is studied in the case when the operator at time derivative is degenerate. The study of the question of non-uniqueness of the solution of the Showalter–Sidorov problem for the Hoff model in the two-dimensional domain is carried out on the basis of the phase space method, which was developed by G.A. Sviridyuk. The conditions of non-uniqueness of the solution in the case when the dimension of the operator kernel at time derivative is equal to 1 or 2 are found. Two approaches for revealing the number of solutions of the Showalter–Sidorov problem in the case when the dimension of the operator kernel at time derivative is equal to 2 are presented. Examples illustrating the non-uniqueness of the solution of the problem on a rectangle are given.

Détail

Keywords: Showalter–Sidorov problem, phase space method, Whitney assemblies, the Hoff equation, non-uniqueness of solutions.
Mots-clés : Sobolev type equations

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