On Liouville’s systems corresponding to self similar solutions of the KellerSegel systems of several populations
Speaker 
Dr. Debabrata Karmakar Technion,
Israel Institute of Technology


When 
Sep 10, 2018
from 04:00 PM to 05:00 PM 
Where  LH 006 
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Abstract: The KellerSegel system in two dimensions represents the evolution of living cells under self attraction and diffusive forces. In its simplest form, it is a conservative driftdiffusion equation for the cell density coupled to an elliptic equation for the chemoattractant concentration. It is known that in two space dimension there is a critical mass βc such that for initial mass β ≤ βc there is global in time existence of solutions while for β > βc finite time blowup occurs. In the subcritical regime (β < βc), the solutions decay as time t goes to infinity, while such solution concentrate, as t goes to infinity for the critical initial mass (β = βc). In the subcritical case, this decay can be resolved by a steady, selfsimilar solution, while no such selfsimilar solution is known to exist in the critical case.
Motivated by the KellerSegel system of several interacting populations, we studied the existence/non existence of steady states in the selfsimilar 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/nonexistence 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).