Solvability of Cauchy problem for a system of first order quasilinear equations with right-hand sides $f_1={a_2}u(t,x) + {b_2}(t)v(t,x),$ $f_2={g_2}v(t,x)$

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.