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Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. (English) Zbl 1439.92042

Summary: We consider a parabolic-elliptic chemotaxis system generalizing \[\begin{aligned} u_t &=\nabla \cdot ((u+1)^{m-1}\nabla u)-\nabla \cdot (u (u+1)^{\sigma -1}\nabla v) \\ 0 &=\Delta v-v+u \end{aligned}\] in bounded smooth domains \(\Omega \subset \mathbb{R}^N\), \(N\geq 3\), 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 \(\sigma \leq 0\)
close to given radially symmetric functions there are many initial data producing unbounded solutions if \(\sigma > m-\frac{N-2}{N}\).
In particular, if \(\sigma\leq 0\) and \(\sigma>m-\frac{N-2}{N}\), there are many initial data evolving into solutions that blow up after infinite time.

MSC:

92C17 Cell movement (chemotaxis, etc.)
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K55 Nonlinear parabolic equations
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References:

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