We study the asymptotic behavior of linear evolution equations of the type ∂tg=Dg+Lg-λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg+Lg. In the case Dg=-∂xg, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=-∂x(xg), 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 ∂tf=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 other words, the linear operator has a spectral gap in the natural L2 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. © 2011 Elsevier Masson SAS.
- Exponential convergence
- Long-time behavior