The maximal solution of the logarithmic fast diffusion equation in two space dimensions

Abstract

We consider the problem $$ \begin{alignat}{2} &u_t = \Delta\log u\quad && x\in\mathbb{R}^2, 0<t<T,\\ &u(x,0)=u_0(x)\quad && x\in\mathbb{R}^2, \end{alignat} $$ with nonnegative and integrable data $u_0(x)$, which appears in Riemannian geometry. We construct a class of maximal solutions of this problem and prove that they generate a semigroup of contractions in $L^1(\mathbb R^2)$, characterized by the property of area loss $$ \dfrac{d}{dt}\int u(x,t)\,dx= -4\pi, $$ which holds until $u(x,t)$ vanishes identically at the time $T=\int u_0(x)\, dx/(4\pi)$. Several constructions of the maximal solutions are proposed and alternative characterizations proved. Sharp uniqueness criteria are obtained.