Invariants and Exponential Rate of Convergence to Steady State in the Renewal Equation

#### P. Gwiazda, B. Perthame

2006, v.12, Issue 2, 413-424

ABSTRACT

We consider the renewal equation (also called McKendrick - VonFoerster) equation that arises as a simple model for structured population dynamics. We use an entropy approach to prove the exponential convergence in long time to the steady state, after renormalization by a damping factor to compensate for the system growth. Our approach, by opposition with the original method of Feller based on Laplace transform, uses the direct variable. It uses new invariants of the equation, to which we systematically associate a condition for the exponential convergence.

Keywords: renewal equation,entropy method,contraction principle,longtime asymptotics