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

. We obtain suﬃcient conditions for the existence and uniqueness of a local solution of the Cauchy problem for a quasilinear system with negative functions of the variable t 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 nonlocal solution of the Cauchy problem for a quasilinear system with negative functions of the variable t .


Introduction
A problem with shift for mixed type equation with two degeneration lines was considered in [8].
For system (1), we consider the following initial conditions: where ϕ 1 (x) and ϕ 2 (x) are given functions. Problem (1), (2) is considered on Ω T = {(t, x) |0 ≤ t ≤ T, x ∈ [0, +∞), T > 0}. In [5], by means of an additional argument method, there were found the conditions of nonlocal solvability of the Cauchy problem for the system subject to the initial conditions (2) on Ω T , where u(t, x) and v(t, x) are unknown functions, Systems (1), (3) appear in various problems in natural sciences. For instance, such systems are applied in models of shallow water [1].
In [5], the existence and uniqueness of a nonlocal solution of the Cauchy problem (3), (2) on Ω T were proved under the following conditions 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 We can avoid setting boundary conditions at x = 0 if By means of the additional argument method, we obtain the following extended characteristic system (see [1]- [7] for details): w 3 (s, t, x) = w 2 (s, s, η 1 ), w 4 (s, t, x) = w 1 (s, s, η 2 ), Unknown functions η i , i = 1, 2, and w j , j = 1, 4, depend not only on t and x, but also on additional argument s. Integrating equations (4)-(7) with respect to the argument s and taking into considerations conditions (8)-(10), we obtain an equivalent system of integral equations: Substituting (11) and (12) into (13)-(15)), we get Lemma 1 Assume that the system of integral equations (16)-(19) has a unique solution w j ∈ C(Γ T ), j = 1, 4, and Proof. From (16) and conditions (18) and (19), we Lemma 2 Let w 1 (s, t, x) and w 2 (s, t, x) satisfy the system of integral equations (16)-(19)). Assume that w 1 (s, t, x), w 2 (s, t, x) together with their firstorder derivatives are continuously differentiable and bounded. Then the pair of functions is a solution to the problem (1)), (2) on Ω T 0 , where T 0 is a constant.
Lemma 2 plays the key role in the additional argument method. It is proved in a standard way (cf., for example, [1]).

Existence of local solution
Let us introduce the following notations: In the next theorem, we provide conditions for the existence of local solution to the problem (1), (2).
Then for each the Cauchy problem (1), (2) has a unique solution which can be found from the system of integral equations (16)-(19).
The proof of Theorem 1 follows from the following lemma, the proof of which can be obtained in the same way it was done in [1]- [7].

Existence of nonlocal solution
In the next theorem, we provide conditions for the existence of nonlocal solution to the problem (1), (2).
Proof. Differentiating (1) with respect to x and denoting we obtain the system of equations: We add following two equations to the system (11)-(15): System (21) can be written in the form As in [2]- [6], one can prove the existence of a continuously differentiable solution to the problem (23). Therefore, As in [5], one can prove that for all t and x on Ω T Since x) ≥ 0 on Ω T , it follows from (13) and (14) that w 1 (s, t, x) ≥ 0, w 2 (s, t, x) ≥ 0 on Γ T . Therefore, u(t, x) = w 1 (t, t, x) ≥ 0, v(t, x) = w 2 (t, t, x) ≥ 0 on Ω T .
Continuing in the similar way, we obtain that functions u(T k , x), v(T k , x) ∈ C 2 ([0, +∞)) such that satisfy the following estimates The second-order derivatives satisfy estimates (27) and (28). As a result, one can extend the solution to any given segment [0, T ] in finitely many steps.
The uniqueness of the solution to the Cauchy problem (1), (2) is proved with the help of estimates similar to those used in the proof of the convergence of successive approximations.