Geodesic
Mixed problem for the two-velocity wave equation with characteristic oblique derivative at the ednpoint of a semibounded string
Matematičeskie zametki, Tome 116 (2024) no. 3, pp. 411-429 Cet article a éte moissonné depuis la source Math-Net.Ru

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We substantiate a formula for a unique and stable smooth solution to the mixed problem for the two-velocity wave equation under a boundary mode with a nonstationary characteristic oblique derivative at the end of a semibounded string. Smooth solutions of order $m$ to this problem are two or more integer $m$ times continuously differentiable solutions to this mixed problem. The characteristic property of the oblique derivative at the end of a semibounded string means that at any moment of time it is directed along the critical characteristic of the wave equation. We derive a criterion for Hadamard well-posedness of the characteristic mixed problem, i.e., necessary and sufficient smoothness requirements for the initial data of this problem and conditions for matching the boundary mode with the initial conditions and the equation. These smoothness requirements and matching conditions ensure the existence, uniqueness, and stability of an $m$ times continuously differentiable solution outside and on the critical characteristic of the equation, respectively.
Keywords: characteristic first oblique derivative, smooth solution, well-posedness criterion, smoothness requirement, matching condition. 
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F. E. Lomovtsev; E. V. Ustilko. Mixed problem for the two-velocity wave equation with characteristic oblique derivative at the ednpoint of a semibounded string. Matematičeskie zametki, Tome 116 (2024) no. 3, pp. 411-429. http://geodesic.mathdoc.fr/item/MZM_2024_116_3_a6/

[1] O. N. Baranovskaya, N. I. Yurchuk, “Smeshannaya zadacha dlya uravneniya kolebaniya struny s zavisyaschei ot vremeni kosoi proizvodnoi v kraevom uslovii”, Differents. uravneniya, 45:8 (2009), 1188–1191 | MR

[2] T. S. Shlapakova, N. I. Yurchuk, “Smeshannaya zadacha dlya uravneniya kolebaniya ogranichennoi struny s zavisyaschei ot vremeni proizvodnoi v kraevom uslovii, napravlennoi po kharakteristike”, Zhurn. Belorus. gos. un-ta. Matem. Informatika, 2 (2013), 84–90

[3] F. E. Lomovtsev, T. S. Tochko, “Gladkie resheniya smeshannoi zadachi dlya prosteishego uravneniya kolebanii poluogranichennoi struny pri kharakteristicheskoi pervoi kosoi proizvodnoi na kontse”, Vesn. Vitseb. un-ta, 3 (120) (2023), 20–36

[4] F. E. Lomovtsev, E. V. Ustilko, “Smeshannaya zadacha dlya odnomernogo volnovogo uravneniya pri kharakteristicheskoi pervoi kosoi proizvodnoi v nestatsionarnom granichnom rezhime dlya gladkikh reshenii”, Vesn. Magil. un-ta, 2 (56) (2020), 21–36

[5] F. E. Lomovtsev, K. A. Spesivtseva, “Smeshannaya zadacha dlya obschego odnomernogo volnovogo uravneniya s kharakteristicheskimi vtorymi proizvodnymi v nestatsionarnom granichnom rezhime”, Matem. zametki, 110:3 (2021), 345–357 | DOI

[6] E. V. Ustilko, F. E. Lomovtsev, “Usloviya soglasovaniya znachenii kharakteristicheskoi kosoi proizvodnoi na kontse struny, nachalnykh dannykh i pravoi chasti volnovogo uravneniya”, Zhurn. Belorus. gos. un-ta. Matem. Informatika, 1 (2020), 30–37 | MR

[7] F. E. Lomovtsev, “Metod korrektirovki probnykh reshenii obschego volnovogo uravneniya v pervoi chetverti ploskosti dlya minimalnoi gladkosti ego pravoi chasti”, Zhurn. Belorus. gos. un-ta. Matem. Informatika, 3 (2017), 38–52 | MR

[8] A. N. Tikhonov, A. A. Samarskii, Uravneniya matematicheskoi fiziki, Nauka, M., 2004 | MR

[9] F. E. Lomovtsev, “Neobkhodimye i dostatochnye usloviya vynuzhdennykh kolebanii poluogranichennoi struny s pervoi kharakteristicheskoi kosoi proizvodnoi v nestatsionarnom granichnom uslovii”, Vest. NAN Belarusi. Ser. fiz.-mat. nauk, 1 (2016), 21–27

[10] F. E. Lomovtsev, V. V. Lysenko, “Nekharakteristicheskaya smeshannaya zadacha dlya odnomernogo volnovogo uravneniya v pervoi chetverti ploskosti pri nestatsionarnykh granichnykh vtorykh proizvodnykh”, Vesn. Vitseb. un-ta, 3 (104) (2019), 5–17

[11] F. E. Lomovtsev, E. V. Ustilko, “Kriterii korrektnosti smeshannoi zadachi dlya obschego uravneniya kolebanii poluogranichennoi struny s nestatsionarnoi kharakteristicheskoi pervoi kosoi proizvodnoi v granichnom uslovii”, Vesn. Vitseb. un-ta, 4 (101) (2018), 18–28

[12] A. P. Khromov, “Raskhodyaschiesya ryady i obobschennaya smeshannaya zadacha dlya volnovogo uravneniya”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Vyp. 21, Saratovskii un-t, Saratov, 2022, 319–324

[13] I. S. Lomov, “Postroenie obobschennogo resheniya smeshannoi zadachi dlya telegrafnogo uravneniya: sekventsialnyi i aksiomaticheskii podkhody”, Differents. uravneniya, 58:11 (2022), 1471–1483 | MR