Semilinear Hyperbolic Systems with Singular Non-Local Boundary Conditions: Reflection of Singularities and Delta Waves

Abstract

In this paper we study initial-boundary value problems for first-order semilinear hyperbolic systems where the boundary conditions are non-local. We focus on situations involving strong singularities, of the Dirac delta type, in the initial data as well as in the boundary conditions. In such cases we prove an existence and uniqueness result in an algebra of generalized functions. Furthermore, we investigate the existence and structure of delta waves, i.e., distributional limits of solutions to the regularized systems. Due to the additional singularities in the boundary data the search for delta waves requires a delicate splitting of the solution into a linearly evolving singular part and a regular part satisfying a nonlinear equation. A new feature in the splitting procedure used here, compared to delta waves in pure initial value problems, is the dependence of the singular part also on part of the regular part due to singularities enetering from the boundary. Finally, we include simple examples where the existence of delta waves breaks down.