The Shock Waves for a Mixed-Type System from Chemotaxis

In this paper, we study the shock waves for a mixed-type system from Chemotaxis. We are concerned with the jump conditions for the left state which is located in the elliptical region and the right state in the hyperbolic region. Under the generalized entropy conditions, we find that there are different shock wave structures for different parameters. To guarantee the uniqueness of the solutions, we obtain the admissible shock waves which satisfy the generalized entropy condition in both parameters. Finally, we construct the Riemann solutions in some solvable regions.