On the Dirichlet Problem for First Order Hyperbolic PDEs on Bounded Domains with Mere Inflow Boundary: Part II Quasi-Linear Equations

Abstract: We study the Dirichlet problem for first order hyperbolic quasi-linear functional PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. While the question of existence of a solution has already been answered in its predecessor, the present paper discusses the uniqueness and continuous dependence on the coefficients of the PDE. Under the assumption that the functional dependence is causal, we are able to derive a contraction principle which is the key to proof uniqueness and continuous dependence. Such a causal functional dependence appears, e.g., in transport based image inpainting.