Goursat problem for quasilinear systems in invariants (Q1902838)

The present article is dealing with the existence and uniqueness question for a classical solution to the Goursat problem on centered wave. A centered wave \(u(t, x)\), \(u_n (t, x)\) is constructed for the system \[ \begin{aligned} u_t+ \lambda (t, x, u, u_n) u_x &= f(t, x, u, u_n),\\ u_{nt}+ \rho t^{\rho-1} a(t, x, u, u_n) u_{nx} &= f_n (t, x, u, u_n), \end{aligned} \] to attain on the characteristic \(x=0\) of the family \(\Gamma_n\) the prescribed values \(u(t, 0)= u_0 (t)\), \(u_n (t, 0)= u_{n0} (t)\) \((0< t\leq \nu_0)\). Here \((t, x)\in \mathbb{R}^2\), \(u_k, u_n: \mathbb{R}^2\to \mathbb{R}\), \(u= (u_k)\), \(f= (f_k)\), \(\lambda= (\lambda_k)\) being a diagonal matrix, \(k= 1, 2, \dots, n-1\), with \(\lambda_n= \rho t^{\rho-1} a\), \(\rho\geq 1\).