Riemann problem for a non-strictly hyperbolic system in chemotaxis

Abstract

The Riemann problem is solved for a system arising in chemotaxis. The system is of mixed-type and transitions from a hyperbolic to an elliptic region. It is genuinely nonlinear in the u-v plane except on the v-axis, where it is linearly degenerate. We have solved the Riemann problem in the physically relevant region up to the boundary of the hyperbolic-elliptic region, which is non-strictly hyperbolic. We also solved the problem on the linearly degenerate region. While solving the Riemann problem, we found classical shock and rarefaction waves in the hyperbolic region and contact discontinuities in the linearly degenerate region.