For the singular Keller-Segel system on a half-line, a unique boundary spike-layer steady state exists and is nonlinearly asymptotically stable for all m≥0, provided χ is sufficiently large when m<1.
Contraction for large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model
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abstract
We consider a hyperbolic-parabolic system arising from a chemotaxis model in angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.
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Boundary spike-layer solutions of the singular Keller-Segel system: existence and stability
For the singular Keller-Segel system on a half-line, a unique boundary spike-layer steady state exists and is nonlinearly asymptotically stable for all m≥0, provided χ is sufficiently large when m<1.