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On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient nonlinearityстатья
Статья опубликована в высокорейтинговом журнале
Статья опубликована в журнале из списка Web of Science и/или Scopus- Авторы: Korpusov M.O., Matveeva A.K.
- Журнал: Mathematical Methods in the Applied Sciences
- Том: 46
- Номер: 2
- Год издания: 2023
- Издательство: John Wiley & Sons Inc.
- Местоположение издательства: United States
- Первая страница: 1574
- Последняя страница: 1630
- DOI: 10.1002/mma.8595
- Аннотация: In this paper, we consider the Cauchy problem for one nonclassical, third-order,partial differential equation with gradient nonlinearity |∇u(x, t)|q. The solutionto this problem is understood in a weak sense. We show that for 1 < q ⩽ 3∕2,there are no local-in-time weak solutions of this problem with initial data u0(x)from the class U, whereas for q > 3∕2, such a solution exists. For 3∕2 < q ⩽2, the Cauchy problem proved not to have global-in-time weak solutions. For3∕2 < q ⩽ 2, we show finite-time blow-up of unique local-in-time weak solutionof the Cauchy problem without dependence from the initial functions from theclass U. As a technique, we obtain Schauder-type estimates for potentials. Weuse them to investigate smoothness of the weak solution to the Cauchy problemfor q = 2 and q ⩾ 4.
- Добавил в систему: Корпусов Максим Олегович