| Details: |
The Keller-Segel system in two dimensions represents the evolution of living cells under self-attraction and
diffusive forces. In its simplest form, it is a conservative drift-diffusion equation for the cell density coupled
to an elliptic equation for the chemo-attractant concentration. It is known that in two space dimension
there is a critical mass $\beta_c$ such that for initial mass $\beta \leq \beta_c$ there is global in time existence of solutions
while for $\beta>\beta_c$ finite time blow-up occurs. In the sub-critical regime $(\beta < \beta_c),$ the solutions decay
as time $t$ goes to infinity, while such solution concentrate, as $t$ goes to infinity
for the critical initial mass $(\beta=\beta_c).$ In the sub-critical case, this decay can be resolved by a steady,
self-similar solution, while no such self-similar solution is known to exist in the critical
case.
Motivated by the Keller-Segel system of several interacting populations, we studied the
existence/non-existence of steady states in the self-similar variables,
when the system has an additional drift for each component decaying in time at the
rate $O(1/\sqrt{t}).$ Such steady states satisfy a modified Liouville's system with a quadratic potential.
In this presentation, we will discuss the conditions for existence/non-existence of solutions of such
Liouville's systems, which, in turn, is related to the existence/non-existence of minimizers to a corresponding free
energy functional (also called the Lyapunov functional) of the system.
This a joint work with Prof. Gershon Wolansky (arXiv:1802.08975).
|