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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

Daniel Balagué 1, , José A. Cañizo 2, and  Pierre Gabriel 3,

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.
Keywords: exponential convergence, growth, entropy, Fragmentation, long-time behavior., eigenvalue problem.
Mathematics Subject Classification: Primary: 35B40, 45C05, 45K05; Secondary: 82D60, 92C3.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Comm. Math. Sci., 7 (2009), 503-510.  Google Scholar

[6]

J. A. J. Metz and O. Diekmann, eds., "The Dynamics of Physiologically Structured Populations," Lecture notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[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.  Google Scholar

[12]

B. Perthame and D. Salort, Distributed elapsed time model for neuron networks,, in preparation., ().   Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Comm. Math. Sci., 7 (2009), 503-510.  Google Scholar

[6]

J. A. J. Metz and O. Diekmann, eds., "The Dynamics of Physiologically Structured Populations," Lecture notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[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.  Google Scholar

[12]

B. Perthame and D. Salort, Distributed elapsed time model for neuron networks,, in preparation., ().   Google Scholar

[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.  Google Scholar

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