Determination of an unknown curve of discontinuity of the solution of a mixed problem for a quasilinear hyperbolic system (Q1076249)

The authors prove local existence and uniqueness of the piecewise smooth solution to the mixed problem for quasilinear hyperbolic system in two independent variables with unknown line of discontinuity (''free interior boundary''). The problem is of the type \[ u_{i,t}+\lambda_ i(x,t,u)u_{i,x}=F_ i(x,t,u),\quad i=1,...,n\quad for\quad 0<t<T,\quad S_ 1(t)<x<S_ 2(t), \] initial conditions for \(<S_ 1(0),S_ 2(0)>\), boundary conditions on curves \(S_ i(t)\) and the following ordinary differential equation for discontinuity line \(x=g(t):\) \(g^{(m)}=H(g,g',...,g^{(m-1)},t,u^-,u^+),\) where limits \(u^-\), \(u^+\) of the solution on the line g satisfy certain additional conditions.