On deficit estimates of the logarithmic Sobolev inequality for the Tsallis entropy via nonlinear diffusion equations

Abstract

We consider functional inequalities for the Tsallis entropy, a one‐parameter extension of the Boltzmann–Shannon entropy. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality is known as an upper bound estimate by the Fisher information. This inequality is related to the dissipative estimate for solutions to the heat equation and plays a crucial role in the study of its asymptotic behavior. In this paper, we focus on the Tsallis entropy associated with nonlinear diffusion equations and derive deficit estimates of the functional inequality for the Tsallis entropy.