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

Maria J. Caceres (1) , José Cañizo (2) , Stéphane Mischler (3)

1 UGR - Departamento de Matematica Aplicada
2 UAB - Departament de Matemàtiques [Barcelona]
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We study the asymptotic behavior of linear evolution equations of the type $\partial_t g = Dg + \LL g - \lambda g$, where $\LL$ is the fragmentation operator, $D$ is a differential operator, and $\lambda$ is the largest eigenvalue of the operator $Dg + \LL g$. 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 $\partial_t f = \LL f$. 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 both cases mentioned above we show these conditions are met for a wide range of fragmentation coefficients, so the exponential convergence holds.

Domaines: Equations aux dérivées partielles [math.AP]

Maria J. Caceres, José Cañizo, Stéphane Mischler. Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations. Journal de Mathématiques Pures et Appliquées, 2011, 96 (4), pp.334-362. ⟨hal-00530425⟩