Today the regular seminar of the “Optimal Control” department was held. The head of the laboratory “Mathematical problems of control” of the Institute of Control Systems of ANAS, professor of ANAS, doctor of mathematics, Ilgar Gurbat oglu Mamedov gave a talk on “On the well-posed solvability of the Neumann problem for a generalized Mangeron equation with non-smooth coefficients”.
We emphasize that in the literature so far, the Neumann problem for the generalized Mangeron equation has been studied only for the case of sufficiently smooth coefficients. In this work, the Neumann problem for the generalized Mangeron equation under more natural conditions on the coefficients of the equation, is investigated for the first time.
In addition, the considered generalized Mangeron equation is a generalization of many model equations of various physical processes (Boussinesq-Love equation, generalized moisture transfer equation, telegraph equation, string vibration equation, heat equation, Aller equation, etc.). Thus, the considered problem has not only theoretical, but also great practical interest.
For the fourth-order generalized Mangeron equation with non-smooth coefficients, defined on a rectangular domain, the Neumann problem with non-classical conditions that do not require matching conditions is considered. While the solution to this problem is sought in the isotropic Sobolev space, it is verified that these terms are equivalent to the classical boundary conditions. The well-posed solvability is constructed based on the integral representation method, by reducing the problem into the equation of integral systems. The well-posed solvability of the Neumann problem for the generalized Mangeron equation is proved by the method of operator equations.
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