Convergence to Traveling Waves in Two Model Systems Related to the Dynamics of Liquid–Vapor Phase Changes

Two prototype systems arising from an earlier model for the dynamics of liquid–vapor phase transitions of retrograde fluids in shock tubes are studied. For the first system, the existence of traveling waves is established. The stability of some traveling waves is proved under some conditions if the traveling wave for the reaction-diffusion part of the system is stable. This shows that the admissibility of traveling waves for the original model cannot be determined by stability consideration alone. This leads to the second prototype system for the original model in which the effect of initiation of nuclei is included. For the second system, the convergence to traveling waves is proved for some initial value problems. This result shows the initiation term has a sizable effect in the long time on the speed of the phase boundary.