| Contents |
| In my talk I will review our recent common results with Christian Stinner related to the fully
parabolic volume filling Keller-Segel model with a probability jump function given by
\[
q(u)=(1+u)^{-\gamma}\;\;\gamma\geq 0.
\]
The most interesting one is a critical mass phenomenon in dimension $2$ which states that for $\gamma\geq 1$
the value of intial mass distinguishes between global-in-time bounded solutions for all the data
with $m < 4(1+\gamma)$ (in the case of radially symmetric solutions the critical value is $8(1+\gamma)$ and
an existence of solutions which become infinite when time goes to $\infty$, however they exist globally
in time, if the initial mass of radially symmetric data exceeds $8(1 +\gamma )$. For $0 < \gamma< 1$ we have
a similar result, the only difference is that it is open whether in the supercritical case solutions
which are known to be infinite blow up in finite time or they exist globally. For the precise results
and proofs see
T.Cieslak, C. Stinner, New critical exponents in a fully parabolic Keller-Segel and applications to volume filling models. Preprint. |
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