Geodesic
Inverse problems for some Sobolev-type mathematical models
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 9 (2016) no. 2, pp. 75-89

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The present article is devoted to the study of mathematical models based the Sobolev-type equations and systems arising in dynamics of a stratified fluid, elasticity theory, hydrodynamics, electrodynamics, etc. Along with a solution we determine an unknown right-hand side and coefficients in a Sobolev-type equations of the forth order. The overdetermination conditions are the values of a solution in a collection of points of a spatial domain. The problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. The existence and uniqueness theorems of solutions for the linear and nonlinear cases are proven. In the linear case the result is global in time and it is local in the nonlinear case. The main spaces in question are the Sobolev spaces.
Keywords: the Sobolev-type model; Sobolev equation; existence and uniqueness theorems; inverse problem; boundary value problem; plasma waves; rotating fluid; the Boussinesq–Love model. 
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     author = {S. G. Pyatkov and S. N. Shergin},
     title = {Inverse problems for some {Sobolev-type} mathematical models},
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S. G. Pyatkov; S. N. Shergin. Inverse problems for some Sobolev-type mathematical models. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 9 (2016) no. 2, pp. 75-89. http://geodesic.mathdoc.fr/item/VYURU_2016_9_2_a6/