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Session 13. Global existence versus blowup in nonlinear parabolic systems
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Critical mass in a volume filling model |
Tomasz Cieślak, IMPAN, Poland
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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
\[
q(u)=(1+u)^{-\gamma},\quad \gamma\geq 0.
\]
The most interesting one is a critical mass phenomenon in dimension
\(2\). It states that for \(\gamma\geq 1\) there exists a value of
intial mass \(m_*\) distinguishing between global-in-time bounded
solutions for initial data with mass \(m<m_*\) (in the case of
radially symmetric solutions the critical value is \(8\pi(1+\gamma)\),
for any data \(m_*=4\pi(1+\gamma)\)) and an existence of solutions
which become infinite when time goes to \(\infty\), however existing
globally in time. The second situation takes place for initial mass
of radially symmetric data exceeding \(8\pi(1+\gamma)\). For \(0<
\gamma<1\) we have a similar result, the only difference is that it
is open whether in the supercritical case solutions blow up in
finite or infinite time.
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