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December 3, 2021

Keller–Segel equation on curved planes

The Keller–Segel equations provide a mathematical model for chemotaxis, that is the organisms (typically bacteria) in the presence of a (chemical) substance. These equations have been intensively studied on R^n with its flat metric, and the most interesting and difficult case is the planar, n = 2. Less is known about solutions in the presence of nonzero curvature. In the talk, I will introduce the Keller–Segel equations in dimension 2, and then briefly recall a few relevant known facts about them. After that I will present my main results. First I prove sharp decay estimates for stationary solutions and prove that such a solution must have mass 8π. Some aspects of this result are novel already in the flat case. Furthermore, using a duality to the “hard” Kazdan–Warner equation on the round sphere, I prove that there are arbitrarily small perturbations of the flat metric on the plane that do not support a stationary solution to the Keller–Segel equations. I will also prove a curved version of the logarithmic Hardy–Littlewood–Sobolev inequality, which I use to prove a result that is complementary to the above ones, as it shows that the functional corresponding to the Keller–Segel equations is bounded from below only when the mass is 8π.

Finally, if time permits, I will present a few results about the nonstationary case, that is a work in progress. In particular, I show long time existence for small masses for certain metrics.