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

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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 and 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), 1482-1518. doi: 10.1137/0523084.
[2]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam, 1978.
[3]	D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135. doi: 10.1016/S0022-247X(03)00457-8.
[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), 171-181.
[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), 289-316. doi: 10.1007/s13160-012-0060-6.
[6]	D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), 99-104. doi: 10.1016/S1631-073X(02)02429-9.
[7]	D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620. doi: 10.1137/080713148.
[8]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, 17, The Clarendon Press Oxford University Press, New York, 1999.
[9]	D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of quasiconvex integrals via the periodic unfolding method, SIAM J. Math. Anal., 37 (2006), 1435-1453 (electronic). doi: 10.1137/040620898.
[10]	A. Damlamian, An elementary introduction to periodic unfolding, Math. Sci. Appl., 24 (2005), 119-136.
[11]	C. Eck, Homogenization of a phase field model for binary mixtures,, Multiscale Model. Simul., 3 (): 1. doi: 10.1137/S1540345903425177.
[12]	J. Elstrodt, Ma$\beta$- und Integrationstheorie, 3rd edition, Springer, 2002.
[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), 305-327. doi: 10.1007/s10492-010-0023-7.
[14]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
[15]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
[16]	T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal., 91 (2012), 1129-1154. doi: 10.1080/00036811.2011.625016.
[17]	B. Fiedler and M. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies, Adv. Differential Equations, 6 (2001), 1377-1408.
[18]	B. Fiedler and M. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms, Asymptot. Anal., 34 (2003), 159-185.
[19]	L. Flodén and M. Olsson, Reiterated homogenization of some linear and nonlinear monotone parabolic operators, Can. Appl. Math. Q., 14 (2006), 149-183.
[20]	A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity, ESAIM Control Optim. Calc. Var., 17 (2011), 1035-1065. doi: 10.1051/cocv/2010036.
[21]	A. Glitzky and R. Hünlich, Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures, Appl. Anal., 66 (1997), 205-226. doi: 10.1080/00036819708840583.
[22]	H. Hanke, Homogenization in gradient plasticity, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651-1684. doi: 10.1142/S0218202511005520.
[23]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981.
[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), 59-94.
[25]	V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, Heidelberg, 1994.
[26]	D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86.
[27]	H. S. Mahato, Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an $H^{1,p}$ Setting, Ph.D thesis, Universit\"at Bremen, 2013.
[28]	V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations, Birkhäuser Boston Inc., Boston, MA, 2006.
[29]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.
[30]	A. Matache and C. Schwab, Two-scale FEM for homogenization problems, Math. Model. Numer. Anal. (M2AN), 36 (2002), 537-572. doi: 10.1051/m2an:2002025.
[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, GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkōtosho, Tokyo, 2010, 443-461.
[32]	A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016.
[33]	A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479-499. doi: 10.3934/dcdss.2013.6.479.
[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), 873-916. doi: 10.1142/S0218202512500637.
[35]	A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Analysis, 39 (2007), 642-668. doi: 10.1137/060672790.
[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), 705-718. doi: 10.1016/j.jmaa.2010.05.056.
[37]	F. Murat and L. Tartar, $H$-convergence, in Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser, Boston, MA, 1997, 21-43.
[38]	J. D. Murray, Mathematical Biology. I. An Introduction, 3rd edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.
[39]	S. Nesenenko, Homogenization in viscoplasticity, SIAM J. Math. Anal., 39 (2007), 236-262. doi: 10.1137/060655092.
[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), 687-720. doi: 10.1137/060665452.
[41]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.
[42]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[43]	J. Persson, Homogenization of monotone parabolic problems with several temporal scales, Appl. Math., 57 (2012), 191-214. doi: 10.1007/s10492-012-0013-z.
[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), 1257-1282. doi: 10.1002/mma.966.
[45]	M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4.
[46]	S. Reichelt, Multi-scale Analysis of Nonlinear Reaction-Diffusion Systems, in preparation, Ph.D thesis, Humboldt-Universität zu Berlin, 2014.
[47]	B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping, SIAM J. Math. Anal., 39 (2008), 1740-1763. doi: 10.1137/060675472.
[48]	B. Schweizer and M. Veneroni, Periodic homogenization of Prandtl-Reuss plasticity with hardening, J. Multiscale Model., 2 (2010), 69-106.
[49]	L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-05195-1.
[50]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.
[51]	A. Visintin, Two-scale convergence of some integral functionals, Calc. Var. Partial Differential Equations, 29 (2007), 239-265. doi: 10.1007/s00526-006-0068-3.
[52]	A. Visintin, Homogenization of a parabolic model of ferromagnetism, J. Differential Equations, 250 (2011), 1521-1552. doi: 10.1016/j.jde.2010.09.016.
[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), 93-107.
[54]	A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397 (electronic). doi: 10.1051/cocv:2006012.
[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), 1363-1401. doi: 10.1017/S0308210506000709.
[56]	J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales, Ann. Mat. Pura Appl. (4), 189 (2010), 357-379. doi: 10.1007/s10231-009-0112-y.

show all references

References:

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.
[2]	A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam, 1978.
[3]	D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, J. Math. Anal. Appl., 286 (2003), 125-135. doi: 10.1016/S0022-247X(03)00457-8.
[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), 171-181.
[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), 289-316. doi: 10.1007/s13160-012-0060-6.
[6]	D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), 99-104. doi: 10.1016/S1631-073X(02)02429-9.
[7]	D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620. doi: 10.1137/080713148.
[8]	D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications, 17, The Clarendon Press Oxford University Press, New York, 1999.
[9]	D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of quasiconvex integrals via the periodic unfolding method, SIAM J. Math. Anal., 37 (2006), 1435-1453 (electronic). doi: 10.1137/040620898.
[10]	A. Damlamian, An elementary introduction to periodic unfolding, Math. Sci. Appl., 24 (2005), 119-136.
[11]	C. Eck, Homogenization of a phase field model for binary mixtures,, Multiscale Model. Simul., 3 (): 1. doi: 10.1137/S1540345903425177.
[12]	J. Elstrodt, Ma$\beta$- und Integrationstheorie, 3rd edition, Springer, 2002.
[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), 305-327. doi: 10.1007/s10492-010-0023-7.
[14]	L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.
[15]	L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
[16]	T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal., 91 (2012), 1129-1154. doi: 10.1080/00036811.2011.625016.
[17]	B. Fiedler and M. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies, Adv. Differential Equations, 6 (2001), 1377-1408.
[18]	B. Fiedler and M. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms, Asymptot. Anal., 34 (2003), 159-185.
[19]	L. Flodén and M. Olsson, Reiterated homogenization of some linear and nonlinear monotone parabolic operators, Can. Appl. Math. Q., 14 (2006), 149-183.
[20]	A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity, ESAIM Control Optim. Calc. Var., 17 (2011), 1035-1065. doi: 10.1051/cocv/2010036.
[21]	A. Glitzky and R. Hünlich, Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures, Appl. Anal., 66 (1997), 205-226. doi: 10.1080/00036819708840583.
[22]	H. Hanke, Homogenization in gradient plasticity, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651-1684. doi: 10.1142/S0218202511005520.
[23]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981.
[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), 59-94.
[25]	V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, Heidelberg, 1994.
[26]	D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86.
[27]	H. S. Mahato, Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an $H^{1,p}$ Setting, Ph.D thesis, Universit\"at Bremen, 2013.
[28]	V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations, Birkhäuser Boston Inc., Boston, MA, 2006.
[29]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.
[30]	A. Matache and C. Schwab, Two-scale FEM for homogenization problems, Math. Model. Numer. Anal. (M2AN), 36 (2002), 537-572. doi: 10.1051/m2an:2002025.
[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, GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkōtosho, Tokyo, 2010, 443-461.
[32]	A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016.
[33]	A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479-499. doi: 10.3934/dcdss.2013.6.479.
[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), 873-916. doi: 10.1142/S0218202512500637.
[35]	A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Analysis, 39 (2007), 642-668. doi: 10.1137/060672790.
[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), 705-718. doi: 10.1016/j.jmaa.2010.05.056.
[37]	F. Murat and L. Tartar, $H$-convergence, in Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser, Boston, MA, 1997, 21-43.
[38]	J. D. Murray, Mathematical Biology. I. An Introduction, 3rd edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.
[39]	S. Nesenenko, Homogenization in viscoplasticity, SIAM J. Math. Anal., 39 (2007), 236-262. doi: 10.1137/060655092.
[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), 687-720. doi: 10.1137/060665452.
[41]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.
[42]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[43]	J. Persson, Homogenization of monotone parabolic problems with several temporal scales, Appl. Math., 57 (2012), 191-214. doi: 10.1007/s10492-012-0013-z.
[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), 1257-1282. doi: 10.1002/mma.966.
[45]	M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4.
[46]	S. Reichelt, Multi-scale Analysis of Nonlinear Reaction-Diffusion Systems, in preparation, Ph.D thesis, Humboldt-Universität zu Berlin, 2014.
[47]	B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping, SIAM J. Math. Anal., 39 (2008), 1740-1763. doi: 10.1137/060675472.
[48]	B. Schweizer and M. Veneroni, Periodic homogenization of Prandtl-Reuss plasticity with hardening, J. Multiscale Model., 2 (2010), 69-106.
[49]	L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-05195-1.
[50]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.
[51]	A. Visintin, Two-scale convergence of some integral functionals, Calc. Var. Partial Differential Equations, 29 (2007), 239-265. doi: 10.1007/s00526-006-0068-3.
[52]	A. Visintin, Homogenization of a parabolic model of ferromagnetism, J. Differential Equations, 250 (2011), 1521-1552. doi: 10.1016/j.jde.2010.09.016.
[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), 93-107.
[54]	A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397 (electronic). doi: 10.1051/cocv:2006012.
[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), 1363-1401. doi: 10.1017/S0308210506000709.
[56]	J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales, Ann. Mat. Pura Appl. (4), 189 (2010), 357-379. doi: 10.1007/s10231-009-0112-y.

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