The Cauchy problem for semilinear hyperbolic systems with discontinuous data (Q1094588)

The author proves short time existence of the solution to the strictly hyperbolic Cauchy problem \[ Lu\equiv u_ t+\sum^{n}_{i=1}A_ i(t,x)u_ i=F(t,x,u),\quad u_{t=0}=g, \] where g is piecewise smooth with jumps only over a hypersurface \(\Gamma \subset R^ n\). The solution is obtained by means of the successive approximation scheme \(Lu_{k+1}=F(t,x,u_ k)\), \(u_{k+1}(t=0)=g\) in the space, roughly speaking, \(L_{\infty}\cap\) conormal distributions (piecewise Sobolev functions cannot be used here because of the lack of stability (solvability of linear system), which is discussed more explicitly).