2 problems

For $N\geq 2$, let $\Omega_0$ be a bounded open subset of $\mathbb{R}^N$ with boundary $\Gamma_0$. Let $u$ solve the heat equation with initial data $\mathbf{1}_{\Omega_0}$, and de…

Phase-transition conjecture. The stationary solution $u(t,x)=Q$ belongs to the boundary of the scattering domain. The ground state is the distinguished stationary threshold separat…