Global stability for the prion equation with general incidence

Pierre Gabriel; Pierre Gabriel

doi:10.3934/mbe.2015.12.789

Mathematical Biosciences and Engineering

2015, Volume 12, Issue 4: 789-801. doi: 10.3934/mbe.2015.12.789

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Global stability for the prion equation with general incidence

Pierre Gabriel ¹

1.
Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue de États-Unis, 78035 Versailles cedex

Received: 01 May 2014 Accepted: 29 June 2018 Published: 01 April 2015
MSC : Primary: 92D25; Secondary: 35B35, 35B40, 35Q92, 45K05.

We consider the so-called prion equation with the general incidence term introduced in [14], and we investigate the stability of the steady states.The method is based on the reduction technique introduced in [11].The argument combines a recent spectral gap result for the growth-fragmentation equation in weighted $L^1$ spaces and the analysis of a nonlinear system of three ordinary differential equations.

Keywords:

Citation: Pierre Gabriel. Global stability for the prion equation with general incidence[J]. Mathematical Biosciences and Engineering, 2015, 12(4): 789-801. doi: 10.3934/mbe.2015.12.789

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Abstract

References

[1]	Kinetic Related Models, 6 (2013), 219-243.
[2]	Comm. Appl. Ind. Math., 1 (2010), 299-308.
[3]	J. Math. Pures Appl., 96 (2011), 334-362.
[4]	J. Biol. Dyn., 4 (2010), 28-42.
[5]	Math. Biosci., 217 (2009), 88-99.
[6]	Comm. Math. Sci., 7 (2009), 839-865.
[7]	Math. Models Methods Appl. Sci., 20 (2010), 757-783.
[8]	J. Math. Anal. Appl., 324 (2006), 98-117.
[9]	Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.
[10]	Math. Comput. Modelling, 53 (2011), 1451-1456.
[11]	Commun. Math. Sci., 10 (2012), 787-820.
[12]	Appl. Math. Lett., 27 (2014), 74-78.
[13]	J. Theoret. Biol., 242 (2006), 598-606.
[14]	SIAM J. Appl. Math., 68 (2007), 154-170.
[15]	Nature, 215 (1967), 1043-1044.
[16]	Cell, 73 (1993), 1055-1058.
[17]	Commun. Math. Sci., 7 (2009), 503-510.
[18]	J. Evol. Equ., 7 (2007), 241-264.
[19]	Biophysical Chemistry, 77 (1999), 139-152.
[20]	J. Math. Pures Appl., 84 (2005), 1235-1260.
[21]	arXiv:1310.7773.
[22]	J. Differential Equations, 210 (2005), 155-177.
[23]	Science, 216 (1982), 136-144.
[24]	Dis. Cont. Dyn. Sys. Ser. B, 6 (2006), 225-235.
[25]	Nature, 437 (2005), 257-261.
[26]	J. Math. Anal. Appl., 324 (2006), 580-603.
[27]	American Mathematical Society, Providence, RI, 1995.
[28]	in Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations, 15 (2007), 387-397.

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