June  2019, 39(6): 2977-3015. doi: 10.3934/dcds.2019124

Bistable reaction equations with doubly nonlinear diffusion

Dipartimento di Matematica "Giuseppe Luigi Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129, Torino, Italia

Dedicated to Professor Juan Luis Vázquez

Received  January 2018 Revised  November 2018 Published  February 2019

Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves that describe the asymptotic behaviour of a larger class of solutions $ 0\leq u(x, t)\leq 1 $ of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In the present setting we consider bistable reaction terms, which present interesting differences w.r.t. the Fisher-KPP framework recently studied in [5]. We find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call "pseudo-linear", i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the "pseudo-linear" case, the travelling waves of the "slow" diffusion setting exhibit free boundaries.

Finally, as a complement of [5], we study the asymptotic behaviour of more general solutions in the presence of a "heterozygote superior" reaction function and doubly nonlinear diffusion ("slow" and "pseudo-linear").

Citation: Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124
References:
[1]

F. AndreuV. Caselles and J. M. Mazón, A Fisher-Kolmogorov equation with finite speed of propagation, J. Differential Equations, 248 (2010), 2528-2561. doi: 10.1016/j.jde.2010.01.005.

[2]

D. G. Aronson, The porous medium equation, in Some Problems in Nonlinear Diffusion (A. Fasano and M. Primicerio, Eds.), Lecture Notes in Mathematics, Springer-Verlag, New York/Berlin, 1224 (1986), 1–46. doi: 10.1007/BFb0072687.

[3]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, in Partial Differential Equations and Related Topic, Lecture Notes in Mathematics, Pub., New York, 446 (1975), 5–49.

[4]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[5]

A. Audrito and J. L. Vázquez, The Fisher-KPP problem with doubly nonlinear diffusion, J. Differential Equations, 263 (2017), 7647-7708. doi: 10.1016/j.jde.2017.08.025.

[6]

A. Audrito and J. L. Vázquez, The Fisher-KPP problem with doubly nonlinear "fast" diffusion, Nonlinear Anal., 157 (2017), 212-248. doi: 10.1016/j.na.2017.03.015.

[7]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022.

[8]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅰ: Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26.

[9]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅱ: General domains, J. Amer. Math. Soc., 23 (2010), 1-34. doi: 10.1090/S0894-0347-09-00633-X.

[10]

H. Berestycki and L. Nirenberg, Traveling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 9 (1992), 497-572. doi: 10.1016/S0294-1449(16)30229-3.

[11]

X. Cabré and J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5.

[12]

J. CalvoJ. CamposV. CasellesO. Sánchez and J. Soler, Pattern formation in a flux limited reaction-diffusion of porous media type, Invent. Math., 206 (2016), 57-108. doi: 10.1007/s00222-016-0649-5.

[13]

N. CalvoJ. I. DíazJ. DuranyE. Schiavi and C. Vázquez, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics, SIAM J. Appl. Math., 63 (2002), 683-707. doi: 10.1137/S0036139901385345.

[14]

J. CamposP. GuerreroO. Sánchez and J. Soler, On the analysis of travelling waves to a nonlinear flux limited reaction–diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 141-155. doi: 10.1016/j.anihpc.2012.07.001.

[15]

J. Campos and J. Soler, Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation, Nonlinear Anal., 137 (2016), 266-290. doi: 10.1016/j.na.2015.12.021.

[16]

A. De Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equation, 93 (1991), 19-61. doi: 10.1016/0022-0396(91)90021-Z.

[17]

E. DiBenedetto, Degenerate Parabolic Equations, Springer Verlag, New York, Series Universitext, 1993. doi: 10.1007/978-1-4612-0895-2.

[18]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224. doi: 10.1090/S0002-9947-1989-0962278-5.

[19]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312. doi: 10.4171/JEMS/198.

[20]

R. EnguicaA. Gavioli and L. Sanchez, A class of singular first order differential equations with applications in reaction-diffusion, Discrete Contin. Dyn. Syst., 33 (2013), 173-191. doi: 10.3934/dcds.2013.33.173.

[21]

J. R. Esteban and J. L. Vázquez, Homogeneous diffusion in $\mathbb{R}$ with power-like nonlinear diffusivity, Arch. Rational Mech. Anal., 103 (1988), 39-80. doi: 10.1007/BF00292920.

[22]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432.

[23]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.

[24]

M. Garrione and M. Strani, Monotone wave fronts for $ (p,q) $-Laplacian driven reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 91-103. doi: 10.3934/dcdss.2019006.

[25]

A. Gárriz, Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour, preprint, arXiv: 1805.10955.

[26]

C. JinJ. Yin and S. Zheng, Traveling waves for a time delayed Newtonian filtration equation, J. Differential Equations, 254 (2013), 1-29. doi: 10.1016/j.jde.2012.08.034.

[27]

A. S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys, 42 (1987), 135-176.

[28]

A. N. KolmogorovI. Petrovsky and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Bull. Univ. Moskou Ser. Internat. Sec. A, 1 (1937), 1-25.

[29]

T. KuusiJ. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J., 61 (2012), 399-430. doi: 10.1512/iumj.2012.61.4513.

[30]

R. LaisterA. T. Peplow and R. E. Beardmore, Finite time extinction in nonlinear diffusion equations, Appl. Math. Lett., 17 (2004), 561-567. doi: 10.1016/S0893-9659(04)90126-7.

[31]

L. S. Leibenzon, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk SSSR, Geography and Geophysics, 9 (1945), 7–10 (Russian).

[32]

J. Li, Cauchy problem and initial trace for a doubly degenerate parabolic equation with strongly nonlinear sources, J. Math. Anal. Appl., 264 (2001), 49-67. doi: 10.1006/jmaa.2001.7553.

[33]

P. Lindqvist, Notes on the $ p $-Laplace equation, University of Jyväskylä, Jyväskylä, 2006.

[34]

H. MatanoF. Punzo and A. Tesei, Front propagation for nonlinear diffusion equations on the hyperbolic space, J. Eur. Math. Soc., 17 (2015), 1199-1227. doi: 10.4171/JEMS/529.

[35]

H. P. McKean, Nagumo's Equation, Adv. Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X.

[36]

C. B. Muratov and X. Zhong, Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations, Discrete Contin. Dyn. Syst., 37 (2017), 915-944. doi: 10.3934/dcds.2017038.

[37]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.

[38]

P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on $ \mathbb{R}^N $, Arch. Ration. Mech. Anal., 199 (2011), 69-97. doi: 10.1007/s00205-010-0316-8.

[39]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.

[40]

J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. Henri Poincaré, 14 (1997), 499-552. doi: 10.1016/S0294-1449(97)80137-0.

[41]

F. Sanchez-Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178.

[42] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.
[43] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007.
[44]

V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math., 75 (1992), 65-80. doi: 10.1007/BF02567072.

[45]

Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791.

show all references

References:
[1]

F. AndreuV. Caselles and J. M. Mazón, A Fisher-Kolmogorov equation with finite speed of propagation, J. Differential Equations, 248 (2010), 2528-2561. doi: 10.1016/j.jde.2010.01.005.

[2]

D. G. Aronson, The porous medium equation, in Some Problems in Nonlinear Diffusion (A. Fasano and M. Primicerio, Eds.), Lecture Notes in Mathematics, Springer-Verlag, New York/Berlin, 1224 (1986), 1–46. doi: 10.1007/BFb0072687.

[3]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation, in Partial Differential Equations and Related Topic, Lecture Notes in Mathematics, Pub., New York, 446 (1975), 5–49.

[4]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.

[5]

A. Audrito and J. L. Vázquez, The Fisher-KPP problem with doubly nonlinear diffusion, J. Differential Equations, 263 (2017), 7647-7708. doi: 10.1016/j.jde.2017.08.025.

[6]

A. Audrito and J. L. Vázquez, The Fisher-KPP problem with doubly nonlinear "fast" diffusion, Nonlinear Anal., 157 (2017), 212-248. doi: 10.1016/j.na.2017.03.015.

[7]

H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55 (2002), 949-1032. doi: 10.1002/cpa.3022.

[8]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅰ: Periodic framework, J. Eur. Math. Soc., 7 (2005), 173-213. doi: 10.4171/JEMS/26.

[9]

H. BerestyckiF. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems. Ⅱ: General domains, J. Amer. Math. Soc., 23 (2010), 1-34. doi: 10.1090/S0894-0347-09-00633-X.

[10]

H. Berestycki and L. Nirenberg, Traveling fronts in cylinders, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 9 (1992), 497-572. doi: 10.1016/S0294-1449(16)30229-3.

[11]

X. Cabré and J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Phys., 320 (2013), 679-722. doi: 10.1007/s00220-013-1682-5.

[12]

J. CalvoJ. CamposV. CasellesO. Sánchez and J. Soler, Pattern formation in a flux limited reaction-diffusion of porous media type, Invent. Math., 206 (2016), 57-108. doi: 10.1007/s00222-016-0649-5.

[13]

N. CalvoJ. I. DíazJ. DuranyE. Schiavi and C. Vázquez, On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics, SIAM J. Appl. Math., 63 (2002), 683-707. doi: 10.1137/S0036139901385345.

[14]

J. CamposP. GuerreroO. Sánchez and J. Soler, On the analysis of travelling waves to a nonlinear flux limited reaction–diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 141-155. doi: 10.1016/j.anihpc.2012.07.001.

[15]

J. Campos and J. Soler, Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation, Nonlinear Anal., 137 (2016), 266-290. doi: 10.1016/j.na.2015.12.021.

[16]

A. De Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equation, 93 (1991), 19-61. doi: 10.1016/0022-0396(91)90021-Z.

[17]

E. DiBenedetto, Degenerate Parabolic Equations, Springer Verlag, New York, Series Universitext, 1993. doi: 10.1007/978-1-4612-0895-2.

[18]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc., 314 (1989), 187-224. doi: 10.1090/S0002-9947-1989-0962278-5.

[19]

Y. Du and H. Matano, Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12 (2010), 279-312. doi: 10.4171/JEMS/198.

[20]

R. EnguicaA. Gavioli and L. Sanchez, A class of singular first order differential equations with applications in reaction-diffusion, Discrete Contin. Dyn. Syst., 33 (2013), 173-191. doi: 10.3934/dcds.2013.33.173.

[21]

J. R. Esteban and J. L. Vázquez, Homogeneous diffusion in $\mathbb{R}$ with power-like nonlinear diffusivity, Arch. Rational Mech. Anal., 103 (1988), 39-80. doi: 10.1007/BF00292920.

[22]

P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361. doi: 10.1007/BF00250432.

[23]

R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.

[24]

M. Garrione and M. Strani, Monotone wave fronts for $ (p,q) $-Laplacian driven reaction-diffusion equations, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 91-103. doi: 10.3934/dcdss.2019006.

[25]

A. Gárriz, Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour, preprint, arXiv: 1805.10955.

[26]

C. JinJ. Yin and S. Zheng, Traveling waves for a time delayed Newtonian filtration equation, J. Differential Equations, 254 (2013), 1-29. doi: 10.1016/j.jde.2012.08.034.

[27]

A. S. Kalashnikov, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Math. Surveys, 42 (1987), 135-176.

[28]

A. N. KolmogorovI. Petrovsky and N. Piscounoff, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Bull. Univ. Moskou Ser. Internat. Sec. A, 1 (1937), 1-25.

[29]

T. KuusiJ. Siljander and J. M. Urbano, Local Hölder continuity for doubly nonlinear parabolic equations, Indiana Univ. Math. J., 61 (2012), 399-430. doi: 10.1512/iumj.2012.61.4513.

[30]

R. LaisterA. T. Peplow and R. E. Beardmore, Finite time extinction in nonlinear diffusion equations, Appl. Math. Lett., 17 (2004), 561-567. doi: 10.1016/S0893-9659(04)90126-7.

[31]

L. S. Leibenzon, General problem of the movement of a compressible fluid in a porous medium, Izv. Akad. Nauk SSSR, Geography and Geophysics, 9 (1945), 7–10 (Russian).

[32]

J. Li, Cauchy problem and initial trace for a doubly degenerate parabolic equation with strongly nonlinear sources, J. Math. Anal. Appl., 264 (2001), 49-67. doi: 10.1006/jmaa.2001.7553.

[33]

P. Lindqvist, Notes on the $ p $-Laplace equation, University of Jyväskylä, Jyväskylä, 2006.

[34]

H. MatanoF. Punzo and A. Tesei, Front propagation for nonlinear diffusion equations on the hyperbolic space, J. Eur. Math. Soc., 17 (2015), 1199-1227. doi: 10.4171/JEMS/529.

[35]

H. P. McKean, Nagumo's Equation, Adv. Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X.

[36]

C. B. Muratov and X. Zhong, Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations, Discrete Contin. Dyn. Syst., 37 (2017), 915-944. doi: 10.3934/dcds.2017038.

[37]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235.

[38]

P. Poláčik, Threshold solutions and sharp transitions for nonautonomous parabolic equations on $ \mathbb{R}^N $, Arch. Ration. Mech. Anal., 199 (2011), 69-97. doi: 10.1007/s00205-010-0316-8.

[39]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045.

[40]

J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders, Ann. Inst. Henri Poincaré, 14 (1997), 499-552. doi: 10.1016/S0294-1449(97)80137-0.

[41]

F. Sanchez-Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equations, J. Math. Biol., 33 (1994), 163-192. doi: 10.1007/BF00160178.

[42] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.
[43] J. L. Vázquez, The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007.
[44]

V. Vespri, On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations, Manuscripta Math., 75 (1992), 65-80. doi: 10.1007/BF02567072.

[45]

Z. Q. Wu, J. N. Zhao, J. X. Yin and H. L. Li, Nonlinear Diffusion Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791.

Figure 1.  The "slow diffusion" area and the "pseudo-linear" line in $ (m, p-1) $-plane. The yellow and orange area are called "fast diffusion" and "very fast diffusion" range, respectively, and they will not be studied in this paper
Figure 2.  Qualitative representation of the reactions of type C and type C', respectively.
Figure 3.  Examples of admissible TWs: Finite and Positive types
Figure 4.  Reactions of type C, range $ \gamma > 0 $, case $ c = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3, 0.7 $. The second case is excluded by the assumption $ \int_0^1u^{m-1}f(u)du > 0 $
Figure 6.  Reactions of type C, range $ \gamma > 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $. The first two pictures show the case $ 0< c < c_{\ast} $, while the others the cases $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 5.  Reactions of type C, range $ \gamma > 0 $. Null isoclines in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $, in the cases $ 0< c < c_0 $ and $ c > c_0 $, respectively
Figure 7.  Reactions of type C, range $ \gamma = 0$ , case $ c = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $.
Figure 8.  Reactions of type C, range $ \gamma = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(u - a) $, $ a = 0.3 $. The first picture shows the case $ 0< c < c_{\ast} $, while the others the cases $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 9.  Reactions of type C', range $ \gamma > 0 $. Null isoclines in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the cases $ 0< c < c_0 $ and $ c > c_0 $, respectively
Figure 10.  Reactions of type C', range $ \gamma > 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the ranges $ 0< c < c_{\ast} $, $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
Figure 11.  Reactions of type C', range $ \gamma = 0 $. Qualitative behaviour of the trajectories in the $ (X, Z) $-plane for $ f(u) = u(1-u)(a - u) $, $ a = 0.3 $, in the ranges $ 0< c < c_{\ast} $, $ c = c_{\ast} $ and $ c > c_{\ast} $, respectively
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