Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations
Forthe equation $$\sum^{m}_{i=t}t^{k_i}U_{x_i,x_i} - U_n + \sum^{m}_{i=t} a_i(x,t)U_{x_i} + b(x, t)u_t + c(x,t)u = 0,$$ $$k_i = \text{const} ≥ 0,\; i = l ..... m, x = (x_1,..., x_m),\; m_>2,$$ we find a many-dimensional analog of the well-known "Gellerstedt condition" $$ a_...
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| Date: | 1994 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
1994
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5607 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511839537856512 |
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| author | Aldashev, S. A. Алдашев, С. А. Алдашев, С. А. |
| author_facet | Aldashev, S. A. Алдашев, С. А. Алдашев, С. А. |
| author_sort | Aldashev, S. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:14:35Z |
| description | Forthe equation
$$\sum^{m}_{i=t}t^{k_i}U_{x_i,x_i} - U_n + \sum^{m}_{i=t} a_i(x,t)U_{x_i} + b(x, t)u_t + c(x,t)u = 0,$$
$$k_i = \text{const} ≥ 0,\; i = l ..... m, x = (x_1,..., x_m),\; m_>2,$$
we find a many-dimensional analog of the well-known "Gellerstedt condition"
$$ a_i(x,t) = O(1)t^{\alpha},\; i = 1,..., m,\, \alpha >\frac{k_1}{2} - 2.$$
We prove that if this condition is satisfied, then the Darboux problems are uniquely solvable. |
| first_indexed | 2026-03-24T03:19:16Z |
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| id | umjimathkievua-article-5607 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:19:16Z |
| publishDate | 1994 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/13/16b317034ebd0bd6795ae0f26c61d513.pdf |
| spelling | umjimathkievua-article-56072020-03-19T09:14:35Z Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations О корректности многомерных задач Дарбу для вырождающихся гиперболических уравнений Aldashev, S. A. Алдашев, С. А. Алдашев, С. А. Forthe equation $$\sum^{m}_{i=t}t^{k_i}U_{x_i,x_i} - U_n + \sum^{m}_{i=t} a_i(x,t)U_{x_i} + b(x, t)u_t + c(x,t)u = 0,$$ $$k_i = \text{const} ≥ 0,\; i = l ..... m, x = (x_1,..., x_m),\; m_>2,$$ we find a many-dimensional analog of the well-known "Gellerstedt condition" $$ a_i(x,t) = O(1)t^{\alpha},\; i = 1,..., m,\, \alpha >\frac{k_1}{2} - 2.$$ We prove that if this condition is satisfied, then the Darboux problems are uniquely solvable. Для рівняння $$\sum^{m}_{i=t}t^{k_i}U_{x_i,x_i} - U_n + \sum^{m}_{i=t} a_i(x,t)U_{x_i} + b(x, t)u_t + c(x,t)u = 0,$$ $$k_i = \text{const} ≥ 0,\; i = l ..... m, x = (x_1,..., x_m),\; m_>2,$$ знайдено багатовимірний аналог відомої умови Геллерстедта $$ a_i(x,t) = O(1)t^{\alpha},\; i = 1,..., m,\, \alpha >\frac{k_1}{2} - 2.$$ при виконанні якої доведені однозначні розв’язуваності задач Дарбу. Встановлені також теореми єдиності розв’язку цих задач. Institute of Mathematics, NAS of Ukraine 1994-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5607 Ukrains’kyi Matematychnyi Zhurnal; Vol. 46 No. 10 (1994); 1304–1311 Український математичний журнал; Том 46 № 10 (1994); 1304–1311 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5607/7899 https://umj.imath.kiev.ua/index.php/umj/article/view/5607/7900 Copyright (c) 1994 Aldashev S. A. |
| spellingShingle | Aldashev, S. A. Алдашев, С. А. Алдашев, С. А. Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations |
| title | Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations |
| title_alt | О корректности многомерных задач Дарбу для вырождающихся гиперболических уравнений |
| title_full | Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations |
| title_fullStr | Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations |
| title_full_unstemmed | Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations |
| title_short | Well-posedness of many-dimensional Darboux problems for degenerating hyperbolic equations |
| title_sort | well-posedness of many-dimensional darboux problems for degenerating hyperbolic equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5607 |
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