摘要：
In this talk we consider the following asymptotic behavior of solutions for regularized equation to some nonlinear hyperbolic balance laws arising from the following topics: the viscous gas flow through discontinuous nozzle, viscous traffic flow model, and the atmosphere hydrodynamic escape model. Through the dynamical system theory approach, we can transfer our steady-state problem into a singularly perturbed system. By analyzing the system in different scales, we are able to construct singular stationary wave solutions. By using geometric singular perturbation theory, we can show there exist true stationary solution for our problem shadowing the singular stationary wave solutions. For some special degenerate singular solutions, we apply more extended theory from geometric singular perturbation to prove that they also persists under the perturbation. Moreover, in the first topics, we also analyze the stability of stationary wave solutions, and in the second topics, we introduce a new entropy condition to ensure the uniqueness of the stationary solutions.