  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.
