# Convex Sobolev inequalities related to unbalanced optimal transport

## TL;DR

This paper investigates the decay of relative entropies in nonlinear drift-diffusion-reaction equations modeled as gradient flows in unbalanced optimal transport, establishing new inequalities without requiring convexity of the functionals.

## Contribution

It introduces novel isoperimetric-type inequalities for controlling relative entropies in gradient flows over Radon measures, even without geodesic convexity.

## Key findings

- Proves exponential decay of relative entropies in the studied equations.
- Establishes new inequalities linking entropies and their productions.
- Extends analysis to non-convex functionals in unbalanced optimal transport.

## Abstract

We study the behaviour of various Lyapunov functionals (relative entropies) along the solutions of a family of nonlinear drift-diffusion-reaction equations coming from statistical mechanics and population dynamics. These equations can be viewed as gradient flows over the space of Radon measures equipped with the Hellinger-Kantorovich distance. The driving functionals of the gradient flows are not assumed to be geodesically convex or semi-convex. We prove new isoperimetric-type functional inequalities, allowing us to control the relative entropies by their productions, which yields the exponential decay of the relative entropies.