Invariant Manifolds of Semilinear Sobolev Type Equations

Статья посвящена обзору результатов автора по исследованию устойчивости полулинейных уравнений соболевского типа с относительно ограниченным оператором. Рассмотрены начально-краевые задачи для уравнений Хоффа, Осколкова нелинейной фильтрации жидкости, Осколкова плоскопараллельного течения жидкости, Бенжамина - Бона - Махони. Эти задачи при подходящем выборе функциональных пространств могут быть рассмотрены как частные случаи задачи Коши для полулинейного уравнения соболевского типа. При исследовании устойчивости мы пользуемся методами фазового пространства, основанными на теории вырожденных (полу)групп операторов, и применяем обобщение классической теоремы Адамара - Перрона. Показано существование устойчивого и неустойчивого инвариантных многообразий, моделируемых устойчивым и неустойчивым инвариантными пространствами линейной части уравнения, в случае, когда фазовое пространство является простым и относительный спектр и мнимая ось не имеют общих точек.

Ключевые слова

Ключевые слова: уравнения соболевского типа; инвариантные многообразия; уравнения Осколкова; уравнение Хоффа; уравнение Бенджамина - Бона - Махони.

Литература

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