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

Abstract

We consider a Cauchy problem for a system of two first order quasilinear differential equations with right-hand sides 1 = 2 (,) + 2 () (,), 2 = 2 (,). 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 1 = 2 (,) + 2 () (,), 2 = 2 (,), under which the solution has the same smoothness in 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 1 = 2 (,) + 2 () (,), 2 = 2 (,) 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 1 = 2 (,) + 2 () (,), 2 = 2 (,). 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 1 = 2 (,) + 2 () (,), 2 = 2 (,) is based on global estimates.

Nonlocal Solvability of the Cauchy Problem for a System with Negative Functions of the Variable $t$

“…Let us introduce the following notations: In the next theorem, we provide conditions for the existence of local solution to the problem (1), (2).…”

“…As in [2]- [6], one can prove the existence of a continuously differentiable solution to the problem (23). Therefore,…”

“…As in [2]- [6], for all t and x, we obtain the following estimates |∂ Owing to the global estimates (24), ( 26)-(28), we can extend the solution to any given segment [0, T ]. For the initial values take u(T 0 , x), v(T 0 , x) ∈ C2 ([0, +∞)) such that…”

“…In the present work, by means of the additional argument method, we determine the nonlocal solvability conditions for the Cauchy problem ( 1), (2) on Ω T in the case when a(t), b(t), c(t), g(t) are continuous and negative functions on [0, T ]. Also, we assume that…”