Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
Mischler, Stéphane; Cañizo, José Alfredo; Caceres, Maria J. (2011), Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathématiques Pures et Appliquées, 96, 4, p. 334-362. http://dx.doi.org/10.1016/j.matpur.2011.01.003
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00530425/fr/
Journal nameJournal de Mathématiques Pures et Appliquées
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Abstract (EN)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.
Subjects / Keywordsself-similar fragmentation equations; partial differential equations; growth-fragmentation equations; symptotic profile; Entropy; Exponential convergence; Self-similarity; Long-time behavior
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