The Riemann problem for a nonlinear non-strictly hyperbolic system arising in biology. (English) Zbl 0579.35061
Some chemotactic bacterial populations are attracted by the gradient of certain chemical substrate, which they metabolize. When the bacteria have depleted the substrate somewhere, this creates a gradient in the concentration of this substrate, which attracts them further. Hence, travelling bands of bacteria are observed. Their macroscopic behavior is described by the Riemann problem for the following system:
\[
u_ t- (u\nu)_ x=0,\quad \nu_ t-u_ x=0,
\]
where u is the concentration of the bacteria. This system is elliptic for \(\nu^ 2+4u\leq 0\), and hyperbolic elsewhere. Each eigenvalue is linearly degenerate along one half of the axis \(\{u=0\}\), and the Hugoniot locus of a point on this axis is unusual. The above problem is solved in the half-plane \(\{\) \(u\geq 0\}\), where the system is strictly hyperbolic, except at the origin.
Reviewer: C.Y.Chan
MSC:
| 35M99 | Partial differential equations of mixed type and mixed-type systems of partial differential equations |
| 35F25 | Initial value problems for nonlinear first-order PDEs |
| 92D25 | Population dynamics (general) |
Keywords:
rarefaction curves; shock curves; contact discontinuities; chemotactic bacterial populations; macroscopic behavior; Riemann problem; Hugoniot locusReferences:
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