Solvability of Cauchy problem for a system of first order quasilinear equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ $f_2={g_2}v(t,x)$
Ufa mathematical journal, Tome 11 (2019) no. 1, pp. 27-41
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We consider a Cauchy problem for a system of two first order quasilinear differential equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ $f_2={g_2}v(t,x)$. 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 $f_1={a_2}u(t,x) + {b_2}(t)v(t,x)$, $f_2={g_2}v(t,x)$, under which the solution has the same smoothness in $x$ 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 $f_1={a_2}u(t,x) + {b_2}(t)v(t,x)$, $f_2={g_2}v(t,x)$ 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 $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ $f_2={g_2}v(t,x)$. 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 $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ $f_2={g_2}v(t,x)$ is based on global estimates.
Keywords:
first order partial differential equations, Cauchy problem, additional argument method, global estimates.
@article{UFA_2019_11_1_a2,
author = {M. V. Dontsova},
title = {Solvability of {Cauchy} problem for a system of first order quasilinear equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ $f_2={g_2}v(t,x)$},
journal = {Ufa mathematical journal},
pages = {27--41},
publisher = {mathdoc},
volume = {11},
number = {1},
year = {2019},
language = {en},
url = {http://geodesic.mathdoc.fr/item/UFA_2019_11_1_a2/}
}
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M. V. Dontsova. Solvability of Cauchy problem for a system of first order quasilinear equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ $f_2={g_2}v(t,x)$. Ufa mathematical journal, Tome 11 (2019) no. 1, pp. 27-41. http://geodesic.mathdoc.fr/item/UFA_2019_11_1_a2/