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

Abstract

We study the asymptotic behavior of linear evolution equations of the type ∂ t g = D g + L g − λ g , where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator D g + L g . In the case D g = − ∂ x g , this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case D g = − ∂ x ( x g ) , it is known that λ = 1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂ t f = L f . 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 other words, the linear operator has a spectral gap in the natural L 2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.