О направлениях исследований уравнений соболевского типа

Статья представляет собой краткий обзор результатов аналитических исследований классов задач для уравнений соболевского типа, полученных научным коллективом в Южно-Уральском государственном университете. В обзор включен ряд результатов по следующем направлениям: исследование разрешимости начальных задач для линейных, полилинейных уравнений соболевского типа и получение условий их устойчивости; исследование разрешимости классов задач для уравнений соболевского типа высокого порядка; исследование разрешимости и единственности начально-конечных задач и задач оптимального управления для уравнений соболевского типа; создание и развитие теории стохастических уравнений соболевского типа; исследование разрешимости задач для уравнений соболевского типа в пространстве К-форм. Получение всех этих результатов базируется на успешном использовании метода фазового пространства и теории вырожденных разрешающих (полу)групп, разработанными профессором Г.А. Свиридюком и развиваемыми его учениками, работающими в университетах нашей страны. Уравнения соболевского типа лежат в основе различных физических, биологических, экономических и других моделей. Краткое изложение совокупности результатов крупного направления современных исследований позволит получить не только актуальное системное представление о нем, но и о дальнейшем его развитии. Статья содержит пять разделов, в библиографию обзора вошли как работы, ставшие базисными для многих последующих результатов, прежде всего численных исследований, так и работы последних лет, которые расширили границы методов теории уравнений соболевского типа.

Ключевые слова

уравнения соболевского типа; метод фазового пространства Г.А. Свиридюка; вырожденные разрешающие (полу)группы; условие Шоуолтера - Сидорова; начально-конечные условия; оптимальное управление.

Литература

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