The nonlocal solvability conditions for a system with constant terms and coefficients of the variable t
DOI:
https://doi.org/10.26577/JMMCS2024-122-02-b3Keywords:
Cauchy problem, quasilinear system, functions, global estimates.Abstract
We consider the Cauchy problem for a system of quasilinear differential equations with constant terms and coefficients of the variable t. We investigate the solvability of the Cauchy problem for a system of quasilinear differential equations with constant terms and coefficients of the variable t using the additional argument method. A theorem on the existence and uniqueness of the local solution of the Cauchy problem for a system of quasilinear differential equations with constant terms and coefficients of the variable $t$ is formulated. We obtain sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem in original coordinates for a system of quasilinear differential equations with constant terms and coefficients of the variable $t.$ A theorem on the existence and uniqueness of the nonlocal solution of the Cauchy problem for a system of quasilinear differential equations with constant terms and coefficients of the variable $t$ is formulated. A theorem on the existence and uniqueness of the nonlocal solution of the Cauchy problem for a system of quasilinear differential equations with constant terms and coefficients of the variable t is proved. The proof of the nonlocal solvability of the Cauchy problem for a system of quasilinear differential equations with constant terms and coefficients of the variable t relies on global estimates.
References
Loitsyanskii L. G., Mechanics of Liquids and Gases, Pergamon Press, 1966.
Rozhdestvenskij B. L., Yanenko N. N., Systems of quasilinear equations and their applications to gas dynamics, Providence, AMS, 1983.
Alekseenko S.N., Dontsova M.V., Pelinovsky D.E., "Global solutions to the shallow water system with a method of an additional argument" , Applicable Analysis, 96:9 (2017): 1444-1465.
Dontsova M.V., "The nonlocal solvability conditions in original coordinates for a system with constant terms" , Quaestiones Mathematicae, 46:8 (2023): 1599–1608.
Dontsova M.V., "Nonlocal Solvability of the Cauchy Problem for a System with Negative Functions of the Variable t" , Armenian Journal of Mathematics, 15:4 (2023): 1–10.
Dontsova M.V., "Solvability of Cauchy problem for a system of first order quasilinear equations with right-hand sides f1 = a2u(t, x) + b2(t)v(t, x), f2 = g2v(t, x)," , Ufa Mathematical Journal, 11:1 (2019): 27–41.
Dontsova M.V., "Solvability of the Cauchy Problem for a Quasilinear System in Original Coordinates" , Journal of Mathematical Sciences, 249:6 (2020): 918–928.
Dontsova M.V., "Nonlocal solvability conditions for Cauchy problem for a system of first order partial differential equations with special right-hand sides" , Ufa Mathematical Journal, 6:4 (2014): 68–80.
Imanaliev M.I., Alekseenko S.N., "To the problem of the existence of a smooth bounded solution to a system of two nonlinear partial differential equations of the first order" , Doklady Mathematics, 64:1 (2001): 10–15
