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We describe a time-dependent functional involving the relative entropy and the P H1 seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functional sheds light on the competition between dissipation and nonlinearity for this equation. It enables us to obtain new results concerning regularity/ blowup issues for the Landau equation with Coulomb potential. [ABSTRACT FROM AUTHOR]

We consider a system of two reaction-diffusion equations coming out of reversible chemistry. When the reaction happens on the totality of the domain, it is known that exponential convergence to equilibrium holds (with explicit rate). We show in this paper that this exponential convergence also holds when the reaction happens only on a given open set of a ball, thanks to an observation estimate deduced by logarithmic convexity. [ABSTRACT FROM AUTHOR]

New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well. For sublinear non-increasing motility functions, convergence to the spatially homogeneous steady state is shown, a dedicated Lyapunov functional being constructed for that purpose. [ABSTRACT FROM AUTHOR]