The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane (Q1906687)

The authors consider the Cauchy problem \[ u_t= \Delta\log u,\quad u(x, 0)= u_0(x).\tag{1} \] The existence of solutions, both global and local in time, as well as the question of uniqueness are investigated. The most striking result is as follows: For every radial \(u_0(x)\in L^1(\mathbb{R}^2)\) there exists a unique maximal solution \(u\in C^\infty(\mathbb{R}^2\times (0, T))\) of the problem (1), characterized by the additional property \[ \int_{\mathbb{R}^2} u(x, t) dx= \int_{\mathbb{R}^2} u(x, 0) dx- 4\pi t, \] and accordingly, the existence time is \(T= \int u(x, 0) dx/4\pi\). The geometrical discussion of the results is also given.