General relative entropy inequality: an illustration on growth models

Abstract

We introduce the notion of General Relative Entropy Inequality for several linear PDEs. This concept extends to equations that are not conservation laws, the notion of relative entropy for conservative parabolic, hyperbolic or integral equations. These are particularly natural in the context of biological applications where birth and death can be described by zeroth order terms. But the concept also has applications to more general growth models as the fragmentation equations. We give several types of applications of the General Relative Entropy Inequality: a priori estimates and existence of solution, long time asymptotic to a steady state, attraction to periodic solutions for periodic forcing.