Orateur
Description
On this talk we will focus on the Keller--Segel system
\begin{equation}
\begin{cases}
\displaystyle\partial_t\rho = \Delta \rho^m - \mathrm{div}\left(\rho \,\nabla u \right)
& \text{in } (0,\infty) \times \mathbb{R}^d,\[6pt]
-\Delta u = \rho
& \text{in } (0,\infty)\times \mathbb{R}^d,
\end{cases}
\end{equation}
for $d \geq 2$ and $m = 2 - \frac{2}{d}$, i.e. the critical exponent. This system exhibits a rich behaviour and its dynamics depend on the initial mass. When the mass is below certain threshold (subcritical mass) there is global-in-time bounded solutions, if we are beyond this threshold (supercritical mass), one can construct solutions with finite time blow-up. Finally, if the mass is critical there exists global-in-time solutions but they are not bounded globally-in-time.
The main goal of the talk is to extend the classical Li--Yau and Aronson--Bénilan estimates in order to cover the Keller--Segel case. We are able to recover the estimate for subcritical and critical mass and, in particular, for a small (computable) mass we also obtain a regularising effect. We follow two strategies: for the small mass case we rely on concavity and harmonic analysis. For the general case of subcritical and critical mass our argument is based on a careful analysis of the subsolutions of the Liouville and the Lane--Emden equations combined with a contradiction argument.
The talk presents joint work with C. Elbar and F. Santambrogio.