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Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system

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  • We consider a parabolic-elliptic chemotaxis system generalizing

    $ \begin{align} & {{u}_{t}}=\nabla \cdot ({{(u+1)}^{m-1}}\nabla u)-\nabla \cdot (u{{(u+1)}^{\sigma -1}}\nabla v) \\ & \ 0=\Delta v-v+u \\ \end{align} $

    in bounded smooth domains $ \Omega \subset \mathbb{R}^N $, $ N\ge 3 $, and with homogeneous Neumann boundary conditions. We show that

    ● solutions are global and bounded if $ \sigma<m-\frac{N-2}{N} $

    ● solutions are global if $ \sigma\le 0 $

    ● close to given radially symmetric functions there are many initial data producing unbounded solutions if $ \sigma>m-\frac{N-2}{N} $.

    In particular, if $ \sigma\le 0 $ and $ \sigma>m-\frac{N-2}{N} $, there are many initial data evolving into solutions that blow up after infinite time.

    Mathematics Subject Classification: Primary: 92C17, 35Q92, 35A01, 35K55.


    \begin{equation} \\ \end{equation}
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