Abstract
The Cauchy problem for a system of first-order quasilinear equations with special right-hand sides is considered. The study of solvability of this system in the original coordinates is based on the method of additional argument. It is proved that the local solution of such system exists and that its smoothness is not lower than the smoothness of the initial conditions. For system of two equations non-local solutions are considered that are continued by finite number of steps from the local solution. Sufficient conditions for the existence of such non-local solution are derived. The proof of the non-local resolvability of the system relies on original global estimates.
Highlights
The study of solvability of this system in the original coordinates is based on the method of additional argument
It is proved that the local solution of such system exists and that its smoothness is not lower than the smoothness of the initial conditions
For system of two equations non-local solutions are considered that are continued by nite number of steps from the local solution
Summary
Ãäå u(t, x), v(t, x) íåèçâåñòíûå ôóíêöèè; a1(t), b1(t), b2(t), c1(t), g1(t) èçâåñòíûå ôóíêöèè; a2, g2 èçâåñòíûå êîíñòàíòû. Äëÿ ñèñòåìû óðàâíåíèé (1.1) îïðåäåëèì íà÷àëüíûå óñëîâèÿ: u(0, x) = φ1(x), v(0, x) = φ2(x).  ñòàòüå [1] îïðåäåëåíû äîñòàòî÷íûå óñëîâèÿ íåëîêàëüíîé ðàçðåøèìîñòè çàäà÷è Êîøè äëÿ ñèñòåìû (1.1), ãäå u(t, x), v(t, x) íåèçâåñòíûå ôóíêöèè, a1(t) = a1, b1(t) = b1, b2(t) = b2, c1(t) = c1, g1(t) = g1; a1, bi, c1, g1, i = 1, 2 èçâåñòíûå ïîëîæèòåëüíûå êîíñòàíòû, a2, g2 èçâåñòíûå êîíñòàíòû, (t, x) ∈ ΩT ñ íà÷àëüíûìè óñëîâèÿìè (1.2).  äàííîé ðàáîòå îïðåäåëåíû äîñòàòî÷íûå óñëîâèÿ ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ëîêàëüíîãî ðåøåíèÿ çàäà÷è Êîøè (1.1)(1.2), ïðè êîòîðûõ ðåøåíèå èìååò òàêóþ æå ãëàäêîñòü ïî x, êàê è íà÷àëüíûå ôóíêöèè çàäà÷è Êîøè è äîñòàòî÷íûå óñëîâèÿ íåëîêàëüíîé ðàçðåøèìîñòè çàäà÷è Êîøè (1.1)(1.2) ñ ïîìîùüþ ìåòîäà äîïîëíèòåëüíîãî àðãóìåíòà
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