The nonlocal solvability conditions for a system with absolute terms for the case of positive coefficients

The Cauchy problem for a system of two quasilinear first order partial differential equations with absolute terms for the case of positive coefficients is considered. The study of the Cauchy’s problem solvability is based on the method of an additional argument, which allows to determine the solution in the original coordinates without involving the inverse function theorem. The existence of the Cauchy’s problem local solution with smoothness not lower than the smoothness of the initial conditions, is proven. Paper determines sufficient conditions for the existence of the Cauchy’s problem nonlocal solution continued by a finite number of steps from the local solution. The proof of the nonlocal resolvability of the Cauchy’s problem for a system of two quasilinear first order partial differential equations with absolute terms for the case of positive coefficients relies on global estimates.