New functional inequalities with applications to the arctan-fast diffusion equation

In this paper, we prove a couple of new nonlinear functional inequalities of Sobolev type akin to the logarithmic Sobolev inequality. In particular, one of the inequalities reads $${\int }_{{\mathbb{S}}^1}\arctan \left(\frac{{\mathrm{\partial }}_{x}u}{u}\right){\mathrm{\partial }}_{x}udx\ge \arctan (\mathrm{\parallel }u(t){\mathrm{\parallel }}_{{\dot{W}}^{1,1}({\mathbb{S}}^1)})\mathrm{\parallel }u(t){\mathrm{\parallel }}_{{\dot{W}}^{1,1}({\mathbb{S}}^1)}\mathrm{.}$$ Then, these inequalities are used in the study of the nonlinear \emph{arctan}-fast diffusion equation $${\mathrm{\partial }}_{t}u-{\mathrm{\partial }}_x\arctan \left(\frac{{\mathrm{\partial }}_{x}u}{u}\right)=0.$$ For this highly nonlinear PDE we establish a number of well-posedness results and qualitative properties.