Nonlinear Fisher information, corresponding functional inequalities and applications

We study the evolution of the nonlinear version of the Fisher information along the quasilinear heat equation. We also provide a nonlinear version of a recent functional inequality (Cieślak–Fuest–Hajduk–Sierżęga, 2024), corresponding to the nonlinear heat equation. Next, applications of our version of nonlinear Fisher information to the 1D critical quasilinear fully parabolic Keller–Segel system are given. In particular, the global existence of solutions to the critical nonlinear diffusion/nonlinear sensitivity 1D fully parabolic Keller–Segel system is obtained for certain type of diffusion. Last, but not least, we also study the version of the Fisher information along the $p$-Laplace equation.