Generalized solutions to quasilinear hyperbolic systems in the second canonical form are investigated. A theorem on existence, uniqueness and continuous dependence upon the boundary data is given. The proof is based on the methods due to L. Cesari and P. Bassanini for systems which are not functional.

We compute and justify rigorous geometric optics expansions for linear hyperbolic boundary value problems that do not satisfy the uniform Lopatinskii condition. We exhibit an amplification phenomenon for the reflection of small high frequency oscillations at the boundary. Our analysis has two important consequences for such hyperbolic boundary value problems. Firstly, we make precise the optimal energy estimate in Sobolev spaces showing that losses of derivatives must occur from the source terms...

In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant $\left\{\right(t,x):t\ge 0,x\ge 0\}$ is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a $C^1$ solution and its $L^1$ stability with certain small initial and boundary data.