Peculiarities of Space Dependent Conservation Laws: Inverse Design and Asymptotics

Recently, results regarding the Inverse Design problem for Conservation Laws and Hamilton-Jacobi equations with x-dependent convex fluxes were obtained in Colombo, Perrollaz, and Sylla (2022), Colombo, Perrollaz, and Sylla (2023). More precisely, characterizations of attainable sets and the set of initial data evolving at a prescribed time into a prescribed profile were obtained. Here, we present an explicit example that underlines deep differences between the x-dependent and x-independent cases. Moreover, we add a detailed analysis of the time asymptotic solution of this example, again underlining differences with the x-independent case.