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

Abstract

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 $${\int }_{{\mathbb{S}}^1}\arctan \left(\frac{{\partial }_{x}u}{u}\right){\partial }_{x}udx\ge \arctan \left(\Vert u(t){\Vert }_{{\dot{W}}^{1,1}({\mathbb{S}}^1)}\right)\Vert u(t){\Vert }_{{\dot{W}}^{1,1}({\mathbb{S}}^1)}.$$ Then, these inequalities are used in the study of the nonlinear \emph{arctan}-fast diffusion equation $${\partial }_{t}u-{\partial }_x\arctan \left(\frac{{\partial }_{x}u}{u}\right)=0.$$ For this highly nonlinear PDE we establish a number of well-posedness results and qualitative properties.