Article

    Ufa Mathematical Journal
    Volume 11, Number 1, pp. 27-41

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


    Dontsova М.V.

    DOI:10.13108/2019-11-1-27

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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.