On the Cauchy problem for quasilinear hyperbolic systems with a retarded argument
The author is concerned with quasilinear hyperbolic systems in the ``second canonic'' (or ``bicharacteristic'') form \[ (1)\quad \sum^{n}_{j=1}A_{ij}(x,y,z(x,y),(z\circ \alpha)(x,y))[\frac{\partial z_ j(x,y)}{dx}+ \] \[ +\sum^{m}_{k=1}\rho_{ik}(x,y,z(x,y),(z\circ \beta)(x,y))\frac{\partial z_ j(x,y)}{\partial y_ k}]= \] \[ =f_ i(x,y,z(x,y),(z\circ \gamma)(x,y)),\quad i=1,...,n, \] where \(z=(z_ 1,...,z_ n)\), \((z\circ \alpha)(x,y)=z(\alpha (x,y))\) and \(\alpha (x,y)=(\alpha_ 0(x,y),\alpha_ 1(x,y),...,\alpha_ m(x,y))\). For the problem obtained by adding to (1) the following initial condition \(z(0,x)=\phi (y)\), \(y\in R^ n\), a theorem on the existence and uniqueness of generalized solutions (in the sense ``almost everywhere'') is proved.
scientific article; zbMATH DE number 4038379
scientific article