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On a chemotaxis model with degenerate diffusion: initial shrinking, eventual smoothness and expanding. (English) Zbl 1442.35484

The following model of chemotaxis, where the porous medium diffusive term and the chemotactic term compete, is considered \[u_t=\nabla\cdot(\phi(u)\nabla u)-\chi\nabla\cdot(u\nabla v),\] \[v_t=\Delta v-\alpha uv.\] This system describes evolution of the density \(u\) of aerobic bacteria and the concentration of oxygen \(v\) in a bounded domain of \({\mathbb R}^N\). Under some assumptions on \(\phi(u)=u^m\), \(m>1\), and the initial data, finite speed of propagation of signal is proved. In some more specific situations either shrinking of supports of solutions or expansion to fill up the whole domain is shown.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35B40 Asymptotic behavior of solutions to PDEs
35K65 Degenerate parabolic equations
92C17 Cell movement (chemotaxis, etc.)
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