Shocks and rarefactions arise in a two-phase model with logistic growth

Multi-phase or mixture models are often used to describe the dynamics of complex fluids. In this work, we use a general transformation to reduce the two-phase system of one spatial and time variable to a system of a single variable. Then we assess the behavior of solutions for the inviscid two-phase model with logistic growth. The growth rate widely impacts the behavior of the solution, producing either shocks or rarefactions. Increasing growth increases the frequency and spread of these waves, and eliminating growth reduces solutions to continuous traveling waves. This analysis generalizes a class of asymptotic/linear results for gel swelling as well as showing the extraordinary richness in the molding framework.