Hyperbolic conservation laws with a moving source

The purpose of this paper is to investigate the wave behavior of hyperbolic conservation laws with a moving source. When the speed of the source is close to one of the characteristic speeds of the system, nonlinear resonance occurs and instability may result. We will study solutions with a single transonic shock wave for a general system u_t + f(u)_x = g(x, u). Suppose that the i^th characteristic speed is close to zero. We propose the following stability criteria: l_i∂g/∂ur_i < 0 for nonlinear stability, l_i∂g/∂ur_i > 0 for nonlinear stability. Here l_i and r_i are the i^th normalized left and right eigenvectors of df/du, respectively. Through the local analysis on the evolution of the speed and strength of the transonic shock wave, the above criterion can be justified. It turns out that the speed of the transonic shock wave is monotone increasing (decreasing) most of the time in the unstable (stable) case. This is shown by introducing a global functional on nonlinear wave interactions, based on the Glimm scheme. In particular, together with the local analysis, we can study the shock speed globally. Such a global approach is absent in the previous works. Using this strategy, we prove the existence of solutions and verify the asymptotic stability (or instability).