This article is cited in 5 scientific papers (total in 5 papers)
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),f1=a2u(t,x)+b2(t)v(t,x),f2=g2v(t,x)f2=g2v(t,x)
Abstract:
We consider a Cauchy problem for a system of two first order quasilinear differential equations with right-hand sides f1=a2u(t,x)+b2(t)v(t,x),f1=a2u(t,x)+b2(t)v(t,x),f2=g2v(t,x)f2=g2v(t,x). We study the solvability of the Cauchy problem on the base of an additional argument method. We obtain the sufficient conditions for the existence and uniqueness
of a local solution to the Cauchy problem in terms of the original coordinates coordinates for a system of two first order quasilinear differential equations with right-hand sides f1=a2u(t,x)+b2(t)v(t,x)f1=a2u(t,x)+b2(t)v(t,x), f2=g2v(t,x)f2=g2v(t,x), under which the solution has the same smoothness in xx as the initial functions in the Cauchy problem does. A theorem on the local existence and uniqueness of a solution to the Cauchy problem is formulated and proved.
The theorem on the local existence and uniqueness of a solution to the Cauchy problem for a system of two first order quasilinear differential equations with right-hand sides f1=a2u(t,x)+b2(t)v(t,x)f1=a2u(t,x)+b2(t)v(t,x), f2=g2v(t,x)f2=g2v(t,x) is proved by the additional argument method. We obtain the sufficient conditions of the existence and uniqueness of a nonlocal solution to the Cauchy problem in terms of the initial coordinates for a system of two first order quasilinear differential equations with right-hand sides f1=a2u(t,x)+b2(t)v(t,x),f1=a2u(t,x)+b2(t)v(t,x),f2=g2v(t,x)f2=g2v(t,x). A theorem on the nonlocal existence and uniqueness of the solution of the Cauchy problem is formulated and proved. The proof of the nonlocal solvability of the Cauchy problem for a system of two quasilinear first order partial differential equations with right-hand sides f1=a2u(t,x)+b2(t)v(t,x),f1=a2u(t,x)+b2(t)v(t,x),f2=g2v(t,x)f2=g2v(t,x) is based on global estimates.
Keywords:
first order partial differential equations, Cauchy problem, additional argument method, global estimates.
Citation:
M. V. Dontsova, “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),f1=a2u(t,x)+b2(t)v(t,x),f2=g2v(t,x)f2=g2v(t,x)”, Ufa Math. J., 11:1 (2019), 27–41
\Bibitem{Don19}
\by M.~V.~Dontsova
\paper Solvability of Cauchy problem for a system of first order quasilinear equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ $f_2={g_2}v(t,x)$
\jour Ufa Math. J.
\yr 2019
\vol 11
\issue 1
\pages 27--41
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\crossref{https://doi.org/10.13108/2019-11-1-27}
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https://doi.org/10.13108/2019-11-1-27
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This publication is cited in the following 5 articles:
M.V. Dontsova, “The nonlocal solvability conditions in original coordinates for a system with constant terms”, Quaestiones Mathematicae, 46:8 (2023), 1599
M. V. Dontsova, “Razreshimost zadachi Koshi dlya odnoi sistemy kvazilineinykh differentsialnykh uravnenii pervogo poryadka”, Vladikavk. matem. zhurn., 23:3 (2021), 64–79
M. V. Dontsova, “Dostatochnye usloviya nelokalnoi razreshimosti sistemy dvukh kvazilineinykh uravnenii pervogo poryadka so svobodnymi chlenami”, Izv. IMI UdGU, 55 (2020), 60–78
M. V. Dontsova, “Solvability of the Cauchy Problem for a Quasilinear System in Original Coordinates”, J Math Sci, 249:6 (2020), 918
M. V. Dontsova, “Usloviya nelokalnoi razreshimosti odnoi sistemy dvukh kvazilineinykh uravnenii pervogo poryadka so svobodnymi chlenami”, Zhurnal SVMO, 21:3 (2019), 317–328
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