# American Institute of Mathematical Sciences

June  2013, 6(2): 219-243. doi: 10.3934/krm.2013.6.219

## Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain 2 School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom 3 Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue de États-Unis, 78035 Versailles cedex, France

Received  October 2012 Revised  November 2012 Published  February 2013

We are concerned with the long-time behavior of the growth-frag-mentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to $0$ and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [1], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.
Citation: Daniel Balagué, José A. Cañizo, Pierre Gabriel. Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. Kinetic and Related Models, 2013, 6 (2) : 219-243. doi: 10.3934/krm.2013.6.219
##### References:
 [1] M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl. (9), 96 (2011), 334-362. doi: 10.1016/j.matpur.2011.01.003. [2] M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783. doi: 10.1142/S021820251000443X. [3] M. Escobedo, S. Mischler and M. Rodríguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125. doi: 10.1016/j.anihpc.2004.06.001. [4] P. Gabriel, "Équations de Transport-Fragmentation et Applications aux Maladies à Prions [Transport-Fragmentation Equations and Applications to Prion Diseases]," Ph.D thesis, Paris, 2011. [5] P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Comm. Math. Sci., 7 (2009), 503-510. [6] J. A. J. Metz and O. Diekmann, eds., "The Dynamics of Physiologically Structured Populations," Lecture notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986. [7] P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16 (2006), 1125-1153. doi: 10.1142/S0218202506001480. [8] P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338 (2004), 697-702. doi: 10.1016/j.crma.2004.03.006. [9] P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235-1260. doi: 10.1016/j.matpur.2005.04.001. [10] B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. [11] B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177. doi: 10.1016/j.jde.2004.10.018. [12] B. Perthame and D. Salort, Distributed elapsed time model for neuron networks, in preparation. [13] R. Wong, "Asymptotic Approximation of Integrals," Corrected reprint of the 1989 original, Classics in Applied Mathematics, 34, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. doi: 10.1137/1.9780898719260.

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##### References:
 [1] M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl. (9), 96 (2011), 334-362. doi: 10.1016/j.matpur.2011.01.003. [2] M. Doumic Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783. doi: 10.1142/S021820251000443X. [3] M. Escobedo, S. Mischler and M. Rodríguez Ricard, On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125. doi: 10.1016/j.anihpc.2004.06.001. [4] P. Gabriel, "Équations de Transport-Fragmentation et Applications aux Maladies à Prions [Transport-Fragmentation Equations and Applications to Prion Diseases]," Ph.D thesis, Paris, 2011. [5] P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Comm. Math. Sci., 7 (2009), 503-510. [6] J. A. J. Metz and O. Diekmann, eds., "The Dynamics of Physiologically Structured Populations," Lecture notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986. [7] P. Michel, Existence of a solution to the cell division eigenproblem, Math. Models Methods Appl. Sci., 16 (2006), 1125-1153. doi: 10.1142/S0218202506001480. [8] P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338 (2004), 697-702. doi: 10.1016/j.crma.2004.03.006. [9] P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235-1260. doi: 10.1016/j.matpur.2005.04.001. [10] B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. [11] B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177. doi: 10.1016/j.jde.2004.10.018. [12] B. Perthame and D. Salort, Distributed elapsed time model for neuron networks, in preparation. [13] R. Wong, "Asymptotic Approximation of Integrals," Corrected reprint of the 1989 original, Classics in Applied Mathematics, 34, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. doi: 10.1137/1.9780898719260.
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