Entropy and Fisher information, where the latter is also known as the entropy production, play essential roles in fields such as physics, biology, and information theory, and they are also powerful tools in mathematics, for instance, in analyzing large time behavior, regularity, and asymptotic properties of solutions. In this talk, we investigate the time evolution of Fisher information for nonlinear diffusion equations on bounded domains with Neumann boundary conditions, extending classical results for the linear heat equation and the porous medium equation on the whole space. In particular, we introduce an alternative formulation of one-dimensional nonlinear Fisher information that reveals its time monotonicity. As an application, the global existence of solutions to the one-dimensional critical quasilinear fully parabolic Keller–Segel system with nonlinear diffusion and nonlinear sensitivity is studied. This is based on joint work with Tomasz Cieślak (IMPAN, Poland) and Kentaro Fujie (Tohoku University, Japan).