Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

TL;DR

This paper investigates the long-term behavior of growth-fragmentation and self-similar fragmentation equations, establishing conditions for exponential convergence to steady states using entropy methods.

Contribution

It provides general conditions under which solutions to these equations converge exponentially fast to their steady states, applicable to a wide range of fragmentation coefficients.

Findings

Exponential convergence to steady states is proven for growth-fragmentation equations.

Entropy-entropy dissipation inequalities are used to establish convergence rates.

Conditions are identified that ensure convergence for various fragmentation coefficients.

Abstract

We study the asymptotic behavior of linear evolution equations of the type \partial_t g = Dg + Lg - \lambda g, where L is the fragmentation operator, D is a differential operator, and {\lambda} is the largest eigenvalue of the operator Dg + Lg. In the case Dg = -\partial_x g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg = -x \partial_x g, it is known that {\lambda} = 2 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation \partial_t f = Lf. By means of entropy-entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In both cases mentioned above we show these conditions are met for a wide range of fragmentation coefficients, so the exponential convergence holds.