Two-scale homogenization of nonlinear
   reaction-diffusion systems with slow diffusion

Alexander Mielke; Sina Reichelt; Marita Thomas

doi:10.3934/nhm.2014.9.353

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June 2014, 9(2): 353-382. doi: 10.3934/nhm.2014.9.353

Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion

Alexander Mielke ^1, , Sina Reichelt ^1, and Marita Thomas ^2,

1.	Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany, Germany
2.	Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin

Received November 2013 Revised April 2014 Published July 2014

Abstract
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We derive a two-scale homogenization limit for reaction-diffusion systems where for some species the diffusion length is of order 1 whereas for the other species the diffusion length is of the order of the periodic microstructure. Thus, in the limit the latter species will display diffusion only on the microscale but not on the macroscale. Because of this missing compactness, the nonlinear coupling through the reaction terms cannot be homogenized but needs to be treated on the two-scale level. In particular, we have to develop new error estimates to derive strong convergence results for passing to the limit.

Keywords: degenerating diffusion, Gronwall estimate., coupled reaction-diffusion equations, folding and unfolding, Two-scale convergence, nonlinear reaction.

Mathematics Subject Classification: 35B25, 35A01, 35K57, 35K65, 35M1.

Citation: Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353

References:

[1]	G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar
[2]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, Studies in Mathematics and its Applications, (1978). Google Scholar
[3]	D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction,, J. Math. Anal. Appl., 286 (2003), 125. doi: 10.1016/S0022-247X(03)00457-8. Google Scholar
[4]	V. Chalupecký, T. Fatima and A. Muntean, Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro mass transfer limit,, Journal of Math-for-Industry, 2B (2010), 171. Google Scholar
[5]	V. Chalupecký and A. Muntean, Semi-discrete finite difference multiscale scheme for a concrete corrosion model: A priori estimates and convergence,, Jpn. J. Ind. Appl. Math., 29 (2012), 289. doi: 10.1007/s13160-012-0060-6. Google Scholar
[6]	D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization,, C. R. Math. Acad. Sci. Paris, 335 (2002), 99. doi: 10.1016/S1631-073X(02)02429-9. Google Scholar
[7]	D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization,, SIAM J. Math. Anal., 40 (2008), 1585. doi: 10.1137/080713148. Google Scholar
[8]	D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford Lecture Series in Mathematics and its Applications, (1999). Google Scholar
[9]	D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of quasiconvex integrals via the periodic unfolding method,, SIAM J. Math. Anal., 37 (2006), 1435. doi: 10.1137/040620898. Google Scholar
[10]	A. Damlamian, An elementary introduction to periodic unfolding,, Math. Sci. Appl., 24 (2005), 119. Google Scholar
[11]	C. Eck, Homogenization of a phase field model for binary mixtures,, Multiscale Model. Simul., 3 (): 1. doi: 10.1137/S1540345903425177. Google Scholar
[12]	J. Elstrodt, Ma$\beta$- und Integrationstheorie,, 3rd edition, (2002). Google Scholar
[13]	E. K. Essel, K. Kuliev, G. Kulieva and L.-E. Persson, Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence,, Appl. Math., 55 (2010), 305. doi: 10.1007/s10492-010-0023-7. Google Scholar
[14]	L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar
[15]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar
[16]	T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion,, Appl. Anal., 91 (2012), 1129. doi: 10.1080/00036811.2011.625016. Google Scholar
[17]	B. Fiedler and M. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies,, Adv. Differential Equations, 6 (2001), 1377. Google Scholar
[18]	B. Fiedler and M. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms,, Asymptot. Anal., 34 (2003), 159. Google Scholar
[19]	L. Flodén and M. Olsson, Reiterated homogenization of some linear and nonlinear monotone parabolic operators,, Can. Appl. Math. Q., 14 (2006), 149. Google Scholar
[20]	A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity,, ESAIM Control Optim. Calc. Var., 17 (2011), 1035. doi: 10.1051/cocv/2010036. Google Scholar
[21]	A. Glitzky and R. Hünlich, Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures,, Appl. Anal., 66 (1997), 205. doi: 10.1080/00036819708840583. Google Scholar
[22]	H. Hanke, Homogenization in gradient plasticity,, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651. doi: 10.1142/S0218202511005520. Google Scholar
[23]	D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar
[24]	U. Hornung, W. Jäger and A. Mikelić, Reactive transport through an array of cells with semi-permeable membranes,, RAIRO Modél. Math. Anal. Numér., 28 (1994), 59. Google Scholar
[25]	V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994). Google Scholar
[26]	D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence,, Int. J. Pure Appl. Math., 2 (2002), 35. Google Scholar
[27]	H. S. Mahato, Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an $H^{1,p}$ Setting,, Ph.D thesis, (2013). Google Scholar
[28]	V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations,, Birkhäuser Boston Inc., (2006). Google Scholar
[29]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar
[30]	A. Matache and C. Schwab, Two-scale FEM for homogenization problems,, Math. Model. Numer. Anal. (M2AN), 36 (2002), 537. doi: 10.1051/m2an:2002025. Google Scholar
[31]	S. A. Meier and A. Muntean, A two-scale reaction-diffusion system: Homogenization and fast-reaction limits,, in Current Advances in Nonlinear Analysis and Related Topics, (2010), 443. Google Scholar
[32]	A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems,, Nonlinearity, 24 (2011), 1329. doi: 10.1088/0951-7715/24/4/016. Google Scholar
[33]	A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions,, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479. doi: 10.3934/dcdss.2013.6.479. Google Scholar
[34]	A. Mielke and E. Rohan, Homogenization of elastic waves in fluid-saturated porous media using the Biot model,, Math. Models Meth. Appl. Sci. (M$^3$AS), 23 (2013), 873. doi: 10.1142/S0218202512500637. Google Scholar
[35]	A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation,, SIAM J. Math. Analysis, 39 (2007), 642. doi: 10.1137/060672790. Google Scholar
[36]	A. Muntean and M. Neuss-Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media,, J. Math. Anal. Appl., 371 (2010), 705. doi: 10.1016/j.jmaa.2010.05.056. Google Scholar
[37]	F. Murat and L. Tartar, $H$-convergence,, in Topics in the mathematical modelling of composite materials, (1997), 21. Google Scholar
[38]	J. D. Murray, Mathematical Biology. I. An Introduction,, 3rd edition, (2002). Google Scholar
[39]	S. Nesenenko, Homogenization in viscoplasticity,, SIAM J. Math. Anal., 39 (2007), 236. doi: 10.1137/060655092. Google Scholar
[40]	M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface,, SIAM J. Math. Anal., 39 (2007), 687. doi: 10.1137/060665452. Google Scholar
[41]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608. doi: 10.1137/0520043. Google Scholar
[42]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[43]	J. Persson, Homogenization of monotone parabolic problems with several temporal scales,, Appl. Math., 57 (2012), 191. doi: 10.1007/s10492-012-0013-z. Google Scholar
[44]	M. A. Peter and M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium,, Math. Meth. Appl. Sci., 31 (2008), 1257. doi: 10.1002/mma.966. Google Scholar
[45]	M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey,, Milan J. Math., 78 (2010), 417. doi: 10.1007/s00032-010-0133-4. Google Scholar
[46]	S. Reichelt, Multi-scale Analysis of Nonlinear Reaction-Diffusion Systems,, in preparation, (2014). Google Scholar
[47]	B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping,, SIAM J. Math. Anal., 39 (2008), 1740. doi: 10.1137/060675472. Google Scholar
[48]	B. Schweizer and M. Veneroni, Periodic homogenization of Prandtl-Reuss plasticity with hardening,, J. Multiscale Model., 2 (2010), 69. Google Scholar
[49]	L. Tartar, The General Theory of Homogenization. A Personalized Introduction,, Lecture Notes of the Unione Matematica Italiana, (2009). doi: 10.1007/978-3-642-05195-1. Google Scholar
[50]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar
[51]	A. Visintin, Two-scale convergence of some integral functionals,, Calc. Var. Partial Differential Equations, 29 (2007), 239. doi: 10.1007/s00526-006-0068-3. Google Scholar
[52]	A. Visintin, Homogenization of a parabolic model of ferromagnetism,, J. Differential Equations, 250 (2011), 1521. doi: 10.1016/j.jde.2010.09.016. Google Scholar
[53]	A. Visintin, Some properties of two-scale convergence,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 15 (2004), 93. Google Scholar
[54]	A. Visintin, Towards a two-scale calculus,, ESAIM Control Optim. Calc. Var., 12 (2006), 371. doi: 10.1051/cocv:2006012. Google Scholar
[55]	A. Visintin, Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity,, Roy. Soc. Edinb. Proc. A, 138 (2008), 1363. doi: 10.1017/S0308210506000709. Google Scholar
[56]	J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales,, Ann. Mat. Pura Appl. (4), 189 (2010), 357. doi: 10.1007/s10231-009-0112-y. Google Scholar

show all references

References:

[1]	G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar
[2]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, Studies in Mathematics and its Applications, (1978). Google Scholar
[3]	D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction,, J. Math. Anal. Appl., 286 (2003), 125. doi: 10.1016/S0022-247X(03)00457-8. Google Scholar
[4]	V. Chalupecký, T. Fatima and A. Muntean, Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro mass transfer limit,, Journal of Math-for-Industry, 2B (2010), 171. Google Scholar
[5]	V. Chalupecký and A. Muntean, Semi-discrete finite difference multiscale scheme for a concrete corrosion model: A priori estimates and convergence,, Jpn. J. Ind. Appl. Math., 29 (2012), 289. doi: 10.1007/s13160-012-0060-6. Google Scholar
[6]	D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization,, C. R. Math. Acad. Sci. Paris, 335 (2002), 99. doi: 10.1016/S1631-073X(02)02429-9. Google Scholar
[7]	D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization,, SIAM J. Math. Anal., 40 (2008), 1585. doi: 10.1137/080713148. Google Scholar
[8]	D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford Lecture Series in Mathematics and its Applications, (1999). Google Scholar
[9]	D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of quasiconvex integrals via the periodic unfolding method,, SIAM J. Math. Anal., 37 (2006), 1435. doi: 10.1137/040620898. Google Scholar
[10]	A. Damlamian, An elementary introduction to periodic unfolding,, Math. Sci. Appl., 24 (2005), 119. Google Scholar
[11]	C. Eck, Homogenization of a phase field model for binary mixtures,, Multiscale Model. Simul., 3 (): 1. doi: 10.1137/S1540345903425177. Google Scholar
[12]	J. Elstrodt, Ma$\beta$- und Integrationstheorie,, 3rd edition, (2002). Google Scholar
[13]	E. K. Essel, K. Kuliev, G. Kulieva and L.-E. Persson, Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence,, Appl. Math., 55 (2010), 305. doi: 10.1007/s10492-010-0023-7. Google Scholar
[14]	L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar
[15]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar
[16]	T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion,, Appl. Anal., 91 (2012), 1129. doi: 10.1080/00036811.2011.625016. Google Scholar
[17]	B. Fiedler and M. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies,, Adv. Differential Equations, 6 (2001), 1377. Google Scholar
[18]	B. Fiedler and M. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms,, Asymptot. Anal., 34 (2003), 159. Google Scholar
[19]	L. Flodén and M. Olsson, Reiterated homogenization of some linear and nonlinear monotone parabolic operators,, Can. Appl. Math. Q., 14 (2006), 149. Google Scholar
[20]	A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity,, ESAIM Control Optim. Calc. Var., 17 (2011), 1035. doi: 10.1051/cocv/2010036. Google Scholar
[21]	A. Glitzky and R. Hünlich, Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures,, Appl. Anal., 66 (1997), 205. doi: 10.1080/00036819708840583. Google Scholar
[22]	H. Hanke, Homogenization in gradient plasticity,, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651. doi: 10.1142/S0218202511005520. Google Scholar
[23]	D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). Google Scholar
[24]	U. Hornung, W. Jäger and A. Mikelić, Reactive transport through an array of cells with semi-permeable membranes,, RAIRO Modél. Math. Anal. Numér., 28 (1994), 59. Google Scholar
[25]	V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994). Google Scholar
[26]	D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence,, Int. J. Pure Appl. Math., 2 (2002), 35. Google Scholar
[27]	H. S. Mahato, Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an $H^{1,p}$ Setting,, Ph.D thesis, (2013). Google Scholar
[28]	V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations,, Birkhäuser Boston Inc., (2006). Google Scholar
[29]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2. Google Scholar
[30]	A. Matache and C. Schwab, Two-scale FEM for homogenization problems,, Math. Model. Numer. Anal. (M2AN), 36 (2002), 537. doi: 10.1051/m2an:2002025. Google Scholar
[31]	S. A. Meier and A. Muntean, A two-scale reaction-diffusion system: Homogenization and fast-reaction limits,, in Current Advances in Nonlinear Analysis and Related Topics, (2010), 443. Google Scholar
[32]	A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems,, Nonlinearity, 24 (2011), 1329. doi: 10.1088/0951-7715/24/4/016. Google Scholar
[33]	A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions,, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479. doi: 10.3934/dcdss.2013.6.479. Google Scholar
[34]	A. Mielke and E. Rohan, Homogenization of elastic waves in fluid-saturated porous media using the Biot model,, Math. Models Meth. Appl. Sci. (M$^3$AS), 23 (2013), 873. doi: 10.1142/S0218202512500637. Google Scholar
[35]	A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation,, SIAM J. Math. Analysis, 39 (2007), 642. doi: 10.1137/060672790. Google Scholar
[36]	A. Muntean and M. Neuss-Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media,, J. Math. Anal. Appl., 371 (2010), 705. doi: 10.1016/j.jmaa.2010.05.056. Google Scholar
[37]	F. Murat and L. Tartar, $H$-convergence,, in Topics in the mathematical modelling of composite materials, (1997), 21. Google Scholar
[38]	J. D. Murray, Mathematical Biology. I. An Introduction,, 3rd edition, (2002). Google Scholar
[39]	S. Nesenenko, Homogenization in viscoplasticity,, SIAM J. Math. Anal., 39 (2007), 236. doi: 10.1137/060655092. Google Scholar
[40]	M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface,, SIAM J. Math. Anal., 39 (2007), 687. doi: 10.1137/060665452. Google Scholar
[41]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608. doi: 10.1137/0520043. Google Scholar
[42]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[43]	J. Persson, Homogenization of monotone parabolic problems with several temporal scales,, Appl. Math., 57 (2012), 191. doi: 10.1007/s10492-012-0013-z. Google Scholar
[44]	M. A. Peter and M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium,, Math. Meth. Appl. Sci., 31 (2008), 1257. doi: 10.1002/mma.966. Google Scholar
[45]	M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey,, Milan J. Math., 78 (2010), 417. doi: 10.1007/s00032-010-0133-4. Google Scholar
[46]	S. Reichelt, Multi-scale Analysis of Nonlinear Reaction-Diffusion Systems,, in preparation, (2014). Google Scholar
[47]	B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping,, SIAM J. Math. Anal., 39 (2008), 1740. doi: 10.1137/060675472. Google Scholar
[48]	B. Schweizer and M. Veneroni, Periodic homogenization of Prandtl-Reuss plasticity with hardening,, J. Multiscale Model., 2 (2010), 69. Google Scholar
[49]	L. Tartar, The General Theory of Homogenization. A Personalized Introduction,, Lecture Notes of the Unione Matematica Italiana, (2009). doi: 10.1007/978-3-642-05195-1. Google Scholar
[50]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar
[51]	A. Visintin, Two-scale convergence of some integral functionals,, Calc. Var. Partial Differential Equations, 29 (2007), 239. doi: 10.1007/s00526-006-0068-3. Google Scholar
[52]	A. Visintin, Homogenization of a parabolic model of ferromagnetism,, J. Differential Equations, 250 (2011), 1521. doi: 10.1016/j.jde.2010.09.016. Google Scholar
[53]	A. Visintin, Some properties of two-scale convergence,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 15 (2004), 93. Google Scholar
[54]	A. Visintin, Towards a two-scale calculus,, ESAIM Control Optim. Calc. Var., 12 (2006), 371. doi: 10.1051/cocv:2006012. Google Scholar
[55]	A. Visintin, Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity,, Roy. Soc. Edinb. Proc. A, 138 (2008), 1363. doi: 10.1017/S0308210506000709. Google Scholar
[56]	J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales,, Ann. Mat. Pura Appl. (4), 189 (2010), 357. doi: 10.1007/s10231-009-0112-y. Google Scholar

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