We study the Cauchy problem for a system of quasilinear equations in the original coordinates by using the additional argument method. We obtain sufficient conditions for the existence and uniqueness of a local solution and show that the solution has the same x-smoothness as the initial function. We also obtain sufficient conditions for the existence and uniqueness of a global solution. Bibliography: 4 titles.
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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), f2 = g2v(t, x),” Ufa Math. J. 11, No. 1, 27–39 (2019).
S. N. Alekseenko, M. V. Dontsova, and P. E. Pelinovsky, “Global solutions to the shallow water system with a method of an additional argument,” Appl. Anal. 96, No. 9, 1444–1465 (2017).
M. I. Imanaliev and S. N. Alekseenko, “On the existence of a smooth bounded solution for a system of two first-order nonlinear partial differential equations,” Dokl. Math. 64, No. 1, 10–15 (2001).
M. V. Dontsova, “Nonlocal solvability conditions for Cauchy rpoblem for a system of first order partial differential equations with special right-hand sides,” Ufa Math. J. 6, No. 4, 66–80 (2014).
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Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 91-100.
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Dontsova, M.V. Solvability of the Cauchy Problem for a Quasilinear System in Original Coordinates. J Math Sci 249, 918–928 (2020). https://doi.org/10.1007/s10958-020-04984-x
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DOI: https://doi.org/10.1007/s10958-020-04984-x