The pressure equation in the fast diffusion range (Q1884107)

The authors study the equation \[ v_t=v\Delta v - \gamma| \nabla v| ^2\qquad \text{in } {\mathbb R}^N\times (0,\infty), \] which is an example of a fully nonlinear parabolic equation in non-divergence form. The equation is not well-understood when \(\gamma>0\). In this paper, the authors concentrate on the range \(\gamma>N/2\). They show that the Cauchy problem is well-posed with arbitrarily large initial data, understood in the sense of Borel \(p\)-trace where \(p=-\gamma<0\), and solutions are classical. The authors also point out why classical approaches do not work.