Thursday, October 15, 2020 — 3:15pm, online talk

**Abstract:** We consider the Cauchy problem for a $2$-space dimensional heat equation with exponential nonlinearity. More precisely, we consider initial data in $H^1(\mathbb R^2)$, and a square-exponential nonlinearity, which is critical in the energy space $H^1(\mathbb R^2)$ in view of the Trudinger-Moser inequality. By means of energy methods, we study the asymptotics of solutions below the ground state energy level. The splitting between blow-up and global existence for low energies is determined by the sign of a suitable functional, and it is related to the corresponding Trudinger-Moser inequality. This is a joint work with Michinori Ishiwata, Bernhard Ruf, and Elide Terraneo.