On some class of initial boundary value problems for second order quasilinear hyperbolic systems (Q2367085)

There are considered some classes of initial boundary value problems for second order quasilinear hyperbolic systems with Neumann and Dirichlet conditions of the form: \[ \begin{aligned} \sum_{I,J=0}^ n a_{IJ} (t, \cdot, D^ 1 u(t)) \partial_ I \partial_ J u(t)+ a_ \Omega (t,\cdot, D^ 1 u(t)) &= f_ \Omega (t) \qquad \text{in } (0,T)\times \Omega,\\ \sum_{i=1}^ n n_ i a_ i (t, \cdot, D^ 1 u(t))+ a_ \Gamma (t,\cdot, D^ 1 u(t)) &= f_ \Gamma (t), \quad u_ 0(t)=0 \quad \text{on } (0,T) \times \Gamma,\\ u(0)= u_ 0, \qquad \partial_ t u(0) &= u_ 1 \quad \text{in } \Omega, \end{aligned} \] where \(\Omega \subset \mathbb{R}^ n\), \(u\) is the vector valued unknown function, \(a_{IJ} \in \mathbb{R}\), \((n_ 1, n_ 2, \dots, n_ n)\) denotes the unit outer normal to \(\Gamma\). For this problem, the author formulates compatibility conditions and defines an iterative process to construct the solution. Using Shibata's method, proposed in some papers quoted in the bibliography, he proves existence and uniqueness of the local smooth solution and the convergence of the iteration scheme.