Registration Number
0213U001028,
0111U000481
, R & D reports
Title
Local, global and asymptotic properties of solutions of singular, spectral and nonclassical problems for elliptic and evolution equations and variational inequalities
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Head
Kovalevsky Alexander Al'bertovich, Shishkov Andrey Evgenievich,
Registration Date
10-01-2013
Organization
Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine
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For nonlinear elliptic second-order equations with right-hand sides from Lebesgue spaces close to L^1, we establish the accuracy of a condition which connect the dimension of the domain, the rate of growth of coefficients of the equations and the exponent of the corresponding Lebesgue space and separate cases of the existence and nonexistence of weak solutions of the Dirichlet problem for the considered equations. Besides, conditions of the nonexistence of weak solutions of the Dirichlet problem for nonlinear elliptic equations of arbitrary even order with right-hand sides from Lebesgue spaces close to L^1 are found. For semilinear parabolic equations with the degenerate absorption potential on different classes of manifolds we obtain exact sufficient conditions of the nonpropagation of supersingularities of solutions along these manifolds. For some important classes of model manifolds, the criterion of the realization such a propagation and, as a result, criterion of the existence of supersingular solutions are established. For the heat equation in the cylinder over the unit circle, the mixed problem is considered with the initial conditions and the general boundary conditions which are invariant under the group of rotations. An explicit formula for the solution in the form of Fourier-Dini series is found, the properties of smothness of the solution are studied and a priori estimates for the operator of the considered problem are established in a weight positive scale of Sobolev spaces. The spectral theory of the one-dimensional Dirac operator with the infinite number of point interactions is constructed.
Product Description
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Бурський Володимир Петрович
Войтович Михайло Володимирович
Голощапова Наталя Іванівна
Зарецька Ганна Олександрівна
Каліта Євген Олександрович
Карабаш Ілля Михайлович
Лесіна Євгенія Вікторівна
Луньов Антон Андрійович
Мірошникова Анастасія Анатоліївна
Маламуд Марк Михайлович
Намлєєва Юлія Валеріївна
Рудакова Ольга Анатоліївна
Степанова Катерина Вадимівна
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2020-04-02
