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April  2022, 21(4): 1139-1155. doi: 10.3934/cpaa.2022013

Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type

1. 

Institute of Applied Mathematics (LS Ⅲ), TU Dortmund University, Vogelpothsweg 87, D-44227 Dortmund, Germany

2. 

AG Biomathematik, TU Dortmund University, Vogelpothsweg 87, D-44227 Dortmund, Germany

* Corresponding author

Received  June 2021 Revised  October 2021 Published  April 2022 Early access  December 2021

We consider a Gierer-Meinhardt system on a surface coupled with a parabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (SIAM J. Appl. Dyn. Syst. 18, 2019). We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The cytosolic diffusion is typically much larger than the lateral diffusion on the membrane. This motivates to a corresponding asymptotic reduction, which consists of a nonlocal system on the membrane. We prove the convergence of solutions of the full system towards unique solutions of the reduction.

Citation: Jan-Phillip Bäcker, Matthias Röger. Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1139-1155. doi: 10.3934/cpaa.2022013
References:
[1]

K. Anguige, Multi-phase Stefan problems for a non-linear one-dimensional model of cell-to-cell adhesion and diffusion, European J. Appl. Math., 21 (2010), 109-136.  doi: 10.1017/S0956792509990167.

[2]

K. Anguige and M. Röger, Global existence for a bulk/surface model for active-transport-induced polarisation in biological cells, J. Math. Anal. Appl., 448 (2017), 213-244.  doi: 10.1016/j.jmaa.2016.10.072.

[3]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, in Grundlehren der Mathematischen Wissenschaften., Springer, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[4]

C. Berding and H. Haken, Pattern formation in morphogenesis. Analytical treatment of the Gierer- Meinhardt model on a sphere, J. Math. Biol., 14 (1982), 133-151.  doi: 10.1007/BF01832840.

[5]

D. BotheM. KöhneS. Maier and J. Saal, Global strong solutions for a class of heterogeneous catalysis models, J. Math. Anal. Appl., 445 (2017), 677-709.  doi: 10.1016/j.jmaa.2016.08.016.

[6]

H. Brezis and P. Mironescu, Gagliardo-Nirenberg inequalities and non-inequalities: the full story, Ann. I. H. Poincare-An., 35 (2018), 1355-1376.  doi: 10.1016/j.anihpc.2017.11.007.

[7]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[8]

K. Disser, Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems, J. Differ. Equ., 269 (2020), 4023-4044.  doi: 10.1016/j.jde.2020.03.021.

[9]

C. M. ElliottT. Ranner and C. Venkataraman, Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics, SIAM J. Math. Anal., 49 (2017), 360-397.  doi: 10.1137/15M1050811.

[10]

K. FellnerE. Latos and B. Q. Tang, Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling, Ann. I. H. Poincare-An., 35 (2018), 643-673.  doi: 10.1016/j.anihpc.2017.07.002.

[11]

H. GarckeJ. KampmannA. Rätz and M. Röger, A coupled surface-Cahn-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci., 26 (2016), 1149-1189.  doi: 10.1142/S0218202516500275.

[12]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Biol. Cyber., 12 (1972), 30-39. 

[13]

D. GomezM. J. Ward and J. Wei, The linear stability of symmetric spike patterns for a bulk-membrane coupled Gierer-Meinhardt model, SIAM J. Appl. Dyn. Syst., 18 (2019), 729-768.  doi: 10.1137/18M1222338.

[14]

J. K. Hale and K. Sakamoto, Shadow systems and attractors in reaction-diffusion equations, Appl. Anal., 32 (1989), 287-303.  doi: 10.1080/00036818908839855.

[15]

S. Hausberg and M. Röger, Well-posedness and fast-diffusion limit for a bulk–surface reaction–diffusion system, Nonlinear Differ. Equ. Appl., 25 (2018), 32 pp. doi: 10.1007/s00030-018-0508-8.

[16]

N. I. Kavallaris and T. Suzuki, On the dynamics of a non-local parabolic equation arising from the gierer–meinhardt system, arXiv: 1605.04083.

[17]

J. P. Keener, Activators and inhibitors in pattern formation, Stud. Appl. Math., 59 (1978), 1-23.  doi: 10.1002/sapm19785911.

[18]

B. N. KholodenkoJ. B. Hoek and H. V. Westerhoff, Why cytoplasmic signalling proteins should be recruited to cell membranes, Trends Cell Biol., 10 (2000), 173-178. 

[19]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1968.

[20]

T. Lamm, Biharmonischer Wärmefluß, Diplomarbeit, Universität Freiburg, Mathematisches Institut, 2001. https://www.math.kit.edu/iana1/~lamm/media/dipl.pdf.

[21]

H. Levine and W. J. Rappel, Membrane-bound Turing patterns, Phys. Rev. E, 72 (2005), 5 pp. doi: 10.1103/PhysRevE.72.061912.

[22]

F. Li and W. M. Ni, On the global existence and finite time blow-up of shadow systems, J. Differ. Equ., 247 (2009), 1762-1776.  doi: 10.1016/j.jde.2009.04.009.

[23]

A. Marciniak-CzochraS. HärtingG. Karch and K. Suzuki, Dynamical spike solutions in a nonlocal model of pattern formation, Nonlinearity, 31 (2018), 1757-1781.  doi: 10.1088/1361-6544/aaa5dc.

[24]

K. Masuda and K. Takahashi, Reaction-diffusion systems in the gierer-meinhardt theory of biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58.  doi: 10.1007/BF03167754.

[25]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discret. Contin. Dynam. Syst. Series S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.

[26]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Am. Math. Soc., 45 (1998), 9-18. 

[27]

We i-Ming Ni and Iz umi Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Am. Math. Soc., 297 (1986), 351-368.  doi: 10.2307/2000473.

[28]

I. L. NovakF. GaoY.-S. ChoiD. ResascoJ. C. Schaff and B. M. Slepchenko, Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology, J. Comput. Phys., 226 (2007), 1271-1290.  doi: 10.1016/j.jcp.2007.05.025.

[29]

F. Rothe, Global solutions of reaction-diffusion systems., volume 1072, Springer, Cham, 1984. doi: 10.1007/BFb0099278.

[30]

T. Roubíček, Nonlinear partial differential equations with applications, volume 153 of International Series of Numerical Mathematics, Birkhäuser/Springer Basel AG, Basel, second edition, 2013. doi: 10.1007/978-3-0348-0513-1.

[31]

A. Rätz and M. Röger, Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks, arXiv: 1305.6172v1.

[32]

V. Sharma and J. Morgan, Global existence of solutions to reaction-diffusion systems with mass transport type boundary conditions, SIAM J. Math. Anal., 48 (2016), 4202-4240.  doi: 10.1137/15M1015145.

[33]

M. E. Taylor, Partial differential equations Ⅲ, in Applied Mathematical Sciences, Springer, New York, second edition, 2011. doi: 10.1007/978-1-4419-7049-7.

[34]

K. E. TeigenX. LiJ. LowengrubF. Wang and A. Voigt, A diffusion-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci., 7 (2009), 1009-1037. 

[35]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72. 

[36]

J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: Uniqueness and spectrum estimates, Eur. J. Appl. Math., 10 (1999), 353-378.  doi: 10.1017/S0956792599003770.

[37]

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, 2006.

show all references

References:
[1]

K. Anguige, Multi-phase Stefan problems for a non-linear one-dimensional model of cell-to-cell adhesion and diffusion, European J. Appl. Math., 21 (2010), 109-136.  doi: 10.1017/S0956792509990167.

[2]

K. Anguige and M. Röger, Global existence for a bulk/surface model for active-transport-induced polarisation in biological cells, J. Math. Anal. Appl., 448 (2017), 213-244.  doi: 10.1016/j.jmaa.2016.10.072.

[3]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, in Grundlehren der Mathematischen Wissenschaften., Springer, New York, 1982. doi: 10.1007/978-1-4612-5734-9.

[4]

C. Berding and H. Haken, Pattern formation in morphogenesis. Analytical treatment of the Gierer- Meinhardt model on a sphere, J. Math. Biol., 14 (1982), 133-151.  doi: 10.1007/BF01832840.

[5]

D. BotheM. KöhneS. Maier and J. Saal, Global strong solutions for a class of heterogeneous catalysis models, J. Math. Anal. Appl., 445 (2017), 677-709.  doi: 10.1016/j.jmaa.2016.08.016.

[6]

H. Brezis and P. Mironescu, Gagliardo-Nirenberg inequalities and non-inequalities: the full story, Ann. I. H. Poincare-An., 35 (2018), 1355-1376.  doi: 10.1016/j.anihpc.2017.11.007.

[7]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[8]

K. Disser, Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems, J. Differ. Equ., 269 (2020), 4023-4044.  doi: 10.1016/j.jde.2020.03.021.

[9]

C. M. ElliottT. Ranner and C. Venkataraman, Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics, SIAM J. Math. Anal., 49 (2017), 360-397.  doi: 10.1137/15M1050811.

[10]

K. FellnerE. Latos and B. Q. Tang, Well-posedness and exponential equilibration of a volume-surface reaction-diffusion system with nonlinear boundary coupling, Ann. I. H. Poincare-An., 35 (2018), 643-673.  doi: 10.1016/j.anihpc.2017.07.002.

[11]

H. GarckeJ. KampmannA. Rätz and M. Röger, A coupled surface-Cahn-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci., 26 (2016), 1149-1189.  doi: 10.1142/S0218202516500275.

[12]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Biol. Cyber., 12 (1972), 30-39. 

[13]

D. GomezM. J. Ward and J. Wei, The linear stability of symmetric spike patterns for a bulk-membrane coupled Gierer-Meinhardt model, SIAM J. Appl. Dyn. Syst., 18 (2019), 729-768.  doi: 10.1137/18M1222338.

[14]

J. K. Hale and K. Sakamoto, Shadow systems and attractors in reaction-diffusion equations, Appl. Anal., 32 (1989), 287-303.  doi: 10.1080/00036818908839855.

[15]

S. Hausberg and M. Röger, Well-posedness and fast-diffusion limit for a bulk–surface reaction–diffusion system, Nonlinear Differ. Equ. Appl., 25 (2018), 32 pp. doi: 10.1007/s00030-018-0508-8.

[16]

N. I. Kavallaris and T. Suzuki, On the dynamics of a non-local parabolic equation arising from the gierer–meinhardt system, arXiv: 1605.04083.

[17]

J. P. Keener, Activators and inhibitors in pattern formation, Stud. Appl. Math., 59 (1978), 1-23.  doi: 10.1002/sapm19785911.

[18]

B. N. KholodenkoJ. B. Hoek and H. V. Westerhoff, Why cytoplasmic signalling proteins should be recruited to cell membranes, Trends Cell Biol., 10 (2000), 173-178. 

[19]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1968.

[20]

T. Lamm, Biharmonischer Wärmefluß, Diplomarbeit, Universität Freiburg, Mathematisches Institut, 2001. https://www.math.kit.edu/iana1/~lamm/media/dipl.pdf.

[21]

H. Levine and W. J. Rappel, Membrane-bound Turing patterns, Phys. Rev. E, 72 (2005), 5 pp. doi: 10.1103/PhysRevE.72.061912.

[22]

F. Li and W. M. Ni, On the global existence and finite time blow-up of shadow systems, J. Differ. Equ., 247 (2009), 1762-1776.  doi: 10.1016/j.jde.2009.04.009.

[23]

A. Marciniak-CzochraS. HärtingG. Karch and K. Suzuki, Dynamical spike solutions in a nonlocal model of pattern formation, Nonlinearity, 31 (2018), 1757-1781.  doi: 10.1088/1361-6544/aaa5dc.

[24]

K. Masuda and K. Takahashi, Reaction-diffusion systems in the gierer-meinhardt theory of biological pattern formation, Japan J. Appl. Math., 4 (1987), 47-58.  doi: 10.1007/BF03167754.

[25]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discret. Contin. Dynam. Syst. Series S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.

[26]

W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Am. Math. Soc., 45 (1998), 9-18. 

[27]

We i-Ming Ni and Iz umi Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Am. Math. Soc., 297 (1986), 351-368.  doi: 10.2307/2000473.

[28]

I. L. NovakF. GaoY.-S. ChoiD. ResascoJ. C. Schaff and B. M. Slepchenko, Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology, J. Comput. Phys., 226 (2007), 1271-1290.  doi: 10.1016/j.jcp.2007.05.025.

[29]

F. Rothe, Global solutions of reaction-diffusion systems., volume 1072, Springer, Cham, 1984. doi: 10.1007/BFb0099278.

[30]

T. Roubíček, Nonlinear partial differential equations with applications, volume 153 of International Series of Numerical Mathematics, Birkhäuser/Springer Basel AG, Basel, second edition, 2013. doi: 10.1007/978-3-0348-0513-1.

[31]

A. Rätz and M. Röger, Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks, arXiv: 1305.6172v1.

[32]

V. Sharma and J. Morgan, Global existence of solutions to reaction-diffusion systems with mass transport type boundary conditions, SIAM J. Math. Anal., 48 (2016), 4202-4240.  doi: 10.1137/15M1015145.

[33]

M. E. Taylor, Partial differential equations Ⅲ, in Applied Mathematical Sciences, Springer, New York, second edition, 2011. doi: 10.1007/978-1-4419-7049-7.

[34]

K. E. TeigenX. LiJ. LowengrubF. Wang and A. Voigt, A diffusion-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface, Commun. Math. Sci., 7 (2009), 1009-1037. 

[35]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72. 

[36]

J. Wei, On single interior spike solutions of the Gierer-Meinhardt system: Uniqueness and spectrum estimates, Eur. J. Appl. Math., 10 (1999), 353-378.  doi: 10.1017/S0956792599003770.

[37]

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, 2006.

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