Speaker: Christos Sourdis (National and Kapodistrian University of Athens)

Title: Liouville type theorems for ancient solutions to the equation $u_t=\Delta u+|u|^{p-1}u$ with critical or supercritical exponent.

Abstract: Firstly, we will provide a survey of Liouville type results for global, ancient or eternal solutions to the nonlinear heat equation $u_t=\Delta u+|u|^{p-1}u$, $p>1$, in the whole space. We will then turn our attention to critical or supercritical exponents $p$ with respect to the Sobolev embedding. In that regime, the steady state problem has a continuum of sign-definite radially symmetric solutions, including singular ones, that decay to zero at infinity. We will particularly focus on the case where $p$ is larger or equal to the so called Joseph-Lundgren exponent, in which case the aforementioned radial steady states form a foliation. Our Liouville type results imply the quasiconvergence as $t\to +\infty$ of a class of solutions to the corresponding initial value problem.