On the first mixed problem in banach spaces for degenerate parabolic equations with changing time direction
О первой смешанной задаче в банаховых пространствах для вырождающихся уравнений с меняющимся направлением времени
On the first mixed problem in banach spaces for degenerate parabolic equations with changing time direction
Статья в журнале
Русский
Библиогр.: с. 56-57 (19 назв.) References: p. 58-59 (19 titles)
517.95
вырождающиеся уравнения; изменение направления времени; функциональные пространства; интегральные тождества; первая смешанная задача; разрешимость; degenerate equations; changing time direction; functional spaces; integral identities; first mixed problem; solvability
Математика
Математические заметки СВФУ. – 2018. – Т. 25, N 4 (100), октябрь-декабрь
С. 45-59
Математические заметки СВФУ
Якутск, Издательство СВФУ
2411-9326 (print)
Журнал включен: РИНЦ
The article is devoted to studying one of the sections of nonclassical differential equations, namely, matters concerned with solvability of parabolic equations with changing second-order time direction. As is known, in ordinary boundary-value problems for strictly parabolic equations, the smoothness of the initial and boundary conditions completely ensures that the solutions belong to the Holder spaces, but in the case of equations with changing time direction, the smoothness of the initial and boundary conditions does not ensure that the solutions belong to these spaces. S.A. Tersenov (for a model parabolic equation with changing time direction) and S.G. Pyatkov (for a more general second-order equation) obtained the necessary and sufficient conditions for solvability of the corresponding mixed problems in Holder spaces. In so doing, they always assumed the initial and boundary conditions being equal to zero. Cases in which the initial and boundary conditions belong to Banach spaces are considered. The functional spaces in which the solutions must be sought are introduced. Relevant a priori estimates, which make it possible to obtain the solvability conditions for these problems, are obtained. The properties of the obtained solutions have been studied. In particular, the equivalence of the Riesz and Littlewood-Paley conditions similar to the conditions for solutions of strictly elliptic and strictly parabolic second order equations is established. A unique solvability of the first mixed problem with boundary and initial functions from the Banach space has been proved.
Петрушко, И. М. О первой смешанной задаче в банаховых пространствах для вырождающихся уравнений с меняющимся направлением времени / И. М. Петрушко, М. И. Петрушко // Математические заметки СВФУ. — 2018. — Т. 25, N 4 (100), октябрь-декабрь. — С. 45-59.
DOI: 10.25587/SVFU.2018.100.20553
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