The Solvability of Mixed Problems for Hyperbolic and Parabolic Equations

Abstract

In the literature on nuclear magnetic resonance in small metal particles. three main lines of approach are found which lead lo qualitztvely differenl predictions for the behaviour of Knight shift and, where available, for the linewidlh. In [his paper it is shown how these differences arise as mnsequences of the different sets of starling hypotheses used in each type of approach. Some characteristics of the nuclear spin-lattice relaxation are discussed as well, and mention is given lo experimental considerations.

A condition for generalized solutions of a parabolic problem for a Petrovskii system to be classical

“…This definition claims the minimal conditions under which the left-hand sides of the problem are calculated with the help of continuous classical derivatives. Note that conditions (b) and (c) allow us to take the traces on S and G in (2) and (3) in the sense of restriction of a continuous function.…”

“…The condition is formulated in terms of belonging of the right-hand sides of the problem to suitable anisotropic Hörmander spaces. The use of the function parameter ϕ allows us to achieve the minimal admissible value of the number parameter s, which is not possible in the framework of Sobolev spaces or Hölder spaces [3,4,22]. As to scalar parabolic problems, conditions of this type are obtained in [10,11].…”

On a nonlocal boundary value problem for the equation fourth-order in partial derivatives

“…Проверка применимости метода разделения переменных для общих линейных гиперболических и параболических уравнений в нормальных областях рассмотрена в работе В.А. Ильина [9].…”

Classical solution of the first mixed problem for the wave equation in the cylyndrical domain

“…Много работ посвящено исследованию корректной постановки задач для гиперболических уравнений с двумя свободными переменными. Для этих задач доказаны критерии корректности или получены достаточные условия для существования единственного классического решения [1][2][3][4][5][6][7]. Однако для уравнений с большим числом независимых переменных таких исследований проведено достаточно мало.…”