January  2022, 42(1): 109-135. doi: 10.3934/dcds.2021109

Global existence of solutions to reaction diffusion systems with mass transport type boundary conditions on an evolving domain

1. 

Department of Mathematics, Indian Institute of Technology Jodhpur, Rajasthan, India

2. 

Department of Mathematics, University of Mumbai, Vidyanagari, Mumbai 400 098, India

* Corresponding author: Jyotshana V. Prajapat

Received  December 2020 Revised  February 2021 Published  January 2022 Early access  October 2021

Fund Project: The first author's research was supported by SEED grant of IIT Jodhpur

We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions on an evolving domain. Using a Lyapunov functional and duality arguments, we establish the existence of component wise non-negative global solutions.

Citation: Vandana Sharma, Jyotshana V. Prajapat. Global existence of solutions to reaction diffusion systems with mass transport type boundary conditions on an evolving domain. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 109-135. doi: 10.3934/dcds.2021109
References:
[1]

S. Abdelmalek and S. Kouachi, Proof of existence of global solutions for m-component reaction-diffusion systems with mixed boundary conditions via the Lyapunov functional method, J. Phys. A, 40 (2007), 12335-12350.  doi: 10.1088/1751-8113/40/41/005.

[2]

I. BarrassE. J. Crampin and P. K. Mainia, Mode transitions in a model reaction-diffusion system driven by domain growth and noise, Bull. Math. Biol., 68 (2006), 981-995.  doi: 10.1007/s11538-006-9106-8.

[3]

E. J. CrampinE. A. Gaffney and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bull. Math. Biol., 61 (1999), 1093-1120.  doi: 10.1006/bulm.1999.0131.

[4]

E. J. CrampinE. A. Gaffney and P. K. Maini, Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model, J. Math. Biol., 44 (2002), 107-128.  doi: 10.1007/s002850100112.

[5]

A. Comanici and M. Golubitsky, Patterns on growing square domains via mode interactions, Dyn. Syst., 23 (2008), 167-206.  doi: 10.1080/14689360801945327.

[6]

J. Ding and S. Li, Blow-up and global solutions for nonlinear reaction–diffusion equations with Neumann boundary conditions, Nonlinear Anal., 68 (2008), 507-514.  doi: 10.1016/j.na.2006.11.016.

[7]

R. Douaifia, S. Abdelmalek and S. Bendoukha, Global existence and asymptotic stablity for a class of coupled reaction-diffusion systems on growing domains, Acta Appl. Math, 171 (2021), 13 pp. doi: 10.1007/s10440-021-00385-7.

[8]

E. B. Fabes and N. M. Riviere, Dirichlet and Neumann problems for the heat equation in $C^1$ cylinders, in Harmonic Analysis in Euclidean Spaces Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978.

[9]

K. FellnerJ. Morgan and B. Q. Tang, Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 635-651.  doi: 10.3934/dcdss.2020334.

[10]

K. FellnerJ. Morgan and B. Q. Tang, Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 281-307.  doi: 10.1016/j.anihpc.2019.09.003.

[11]

A. HahnK. Held and L. Tobiska, Modelling of surfactant concentration in a coupled bulk surface problem, PAMM Proc. Appl. Math. Mech, 14 (2014), 525-526.  doi: 10.1002/pamm.201410250.

[12]

S. L. HollisR. H. MartinJr . and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744-761.  doi: 10.1137/0518057.

[13]

S. Kondo and R. Asai, A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.  doi: 10.1038/376765a0.

[14]

A. L. KrauseM. A. Ellis and R. A. Van Gorder, Influence of curvature, growth, and anisotropy on the evolution of turing patterns on growing manifolds, Bull. Math. Biol., 81 (2019), 759-799.  doi: 10.1007/s11538-018-0535-y.

[15]

P. M. KulesaG. C. CruywagenS. R. LubkinP. K. MainiJ. SneydM. W. J. Ferguson and J. D. Murray, On a model mechanism for the spatial pattering of teeth primordia in the alligator, J. Theoret. Biol., 180 (1996), 287-296. 

[16]

M. Labadie, The stabilizing effect of growth on pattern formation, Preprint, (2008).

[17] O.-A. Ladyzhenskaia and N.-N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London, 1968. 
[18]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

[19]

A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction–diffusion systems on fixed and growing domains, J. Comput. Phys., 214 (2006), 239-263.  doi: 10.1016/j.jcp.2005.09.012.

[20]

A. Madzvamuse and A. H. Chung, Analysis and simulations of coupled bulk-surface reaction-diffusion systems on exponentially evolving volumes, Math. Model. Nat. Phenom., 11 (2016), 4-32.  doi: 10.1051/mmnp/201611502.

[21]

A. MadzvamuseE. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains, J. Math. Biol, 61 (2010), 133-164.  doi: 10.1007/s00285-009-0293-4.

[22]

A. Madzvamuse and P. K. Maini, Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains, J. Comput. Phys., 225 (2007), 100-119.  doi: 10.1016/j.jcp.2006.11.022.

[23]

J. Morgan, Global existence for semilinear parabolic systems, SIAM J. Math. Anal, 20 (1989), 1128-1144.  doi: 10.1137/0520075.

[24]

J. Morgan and V. Sharma, Global existence of solutions to volume-surface reaction diffusion systems with dynamic boundary conditions, Differential Integral Equations, 33 (2020), 113-139. 

[25]

R. G. PlazaF. Sànchez-GarduñoP. PadillaR. A. Barrio and P. K. Maini, The effect of growth and curvature on pattern formation, J. Dynam. Differential Equations, 16 (2004), 1093-1121.  doi: 10.1007/s10884-004-7834-8.

[26]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106.  doi: 10.1137/S0036144599359735.

[27]

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.

[28]

A. Rätz and M. Röger, Turing instabilities in a mathematical model for signaling networks, J. Math. Biol., 65 (2012), 1215-1244.  doi: 10.1007/s00285-011-0495-4.

[29]

A. Rätz and M. Röger, Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks, Nonlinearity, 27 (2014), 1805-1827. 

[30]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, Vol. 1072, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.

[31]

V. Sharma, Global existence and uniform estimates for solutions to reaction-diffusion systems with mass transport type of boundary conditions, Comunication on Pure and Applied Analysis, (2020)

[32]

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

[33]

V. Sharma and J. Morgan, Uniform bounds for solutions to volume-surface reaction diffusion systems, Differential Integral Equations, 30 (2017), 423-442. 

[34]

V. Sharma and J. V. Prajapat, Global existence of solution to volume surface reaction diffusion system with evolving domain, work-in-progress.

[35]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B. Biol. Sci., 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[36]

C. VenkataramanO. Lakkis and A. Madzvamuse, Global existence for semilinear reaction–diffusion systems on evolving domains, Journal of Mathematical Biology, 64 (2012), 41-67.  doi: 10.1007/s00285-011-0404-x.

show all references

References:
[1]

S. Abdelmalek and S. Kouachi, Proof of existence of global solutions for m-component reaction-diffusion systems with mixed boundary conditions via the Lyapunov functional method, J. Phys. A, 40 (2007), 12335-12350.  doi: 10.1088/1751-8113/40/41/005.

[2]

I. BarrassE. J. Crampin and P. K. Mainia, Mode transitions in a model reaction-diffusion system driven by domain growth and noise, Bull. Math. Biol., 68 (2006), 981-995.  doi: 10.1007/s11538-006-9106-8.

[3]

E. J. CrampinE. A. Gaffney and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bull. Math. Biol., 61 (1999), 1093-1120.  doi: 10.1006/bulm.1999.0131.

[4]

E. J. CrampinE. A. Gaffney and P. K. Maini, Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model, J. Math. Biol., 44 (2002), 107-128.  doi: 10.1007/s002850100112.

[5]

A. Comanici and M. Golubitsky, Patterns on growing square domains via mode interactions, Dyn. Syst., 23 (2008), 167-206.  doi: 10.1080/14689360801945327.

[6]

J. Ding and S. Li, Blow-up and global solutions for nonlinear reaction–diffusion equations with Neumann boundary conditions, Nonlinear Anal., 68 (2008), 507-514.  doi: 10.1016/j.na.2006.11.016.

[7]

R. Douaifia, S. Abdelmalek and S. Bendoukha, Global existence and asymptotic stablity for a class of coupled reaction-diffusion systems on growing domains, Acta Appl. Math, 171 (2021), 13 pp. doi: 10.1007/s10440-021-00385-7.

[8]

E. B. Fabes and N. M. Riviere, Dirichlet and Neumann problems for the heat equation in $C^1$ cylinders, in Harmonic Analysis in Euclidean Spaces Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978.

[9]

K. FellnerJ. Morgan and B. Q. Tang, Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 635-651.  doi: 10.3934/dcdss.2020334.

[10]

K. FellnerJ. Morgan and B. Q. Tang, Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 281-307.  doi: 10.1016/j.anihpc.2019.09.003.

[11]

A. HahnK. Held and L. Tobiska, Modelling of surfactant concentration in a coupled bulk surface problem, PAMM Proc. Appl. Math. Mech, 14 (2014), 525-526.  doi: 10.1002/pamm.201410250.

[12]

S. L. HollisR. H. MartinJr . and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744-761.  doi: 10.1137/0518057.

[13]

S. Kondo and R. Asai, A reaction–diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768.  doi: 10.1038/376765a0.

[14]

A. L. KrauseM. A. Ellis and R. A. Van Gorder, Influence of curvature, growth, and anisotropy on the evolution of turing patterns on growing manifolds, Bull. Math. Biol., 81 (2019), 759-799.  doi: 10.1007/s11538-018-0535-y.

[15]

P. M. KulesaG. C. CruywagenS. R. LubkinP. K. MainiJ. SneydM. W. J. Ferguson and J. D. Murray, On a model mechanism for the spatial pattering of teeth primordia in the alligator, J. Theoret. Biol., 180 (1996), 287-296. 

[16]

M. Labadie, The stabilizing effect of growth on pattern formation, Preprint, (2008).

[17] O.-A. Ladyzhenskaia and N.-N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London, 1968. 
[18]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, (Russian) Translated from the Russian by S. Smith Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.

[19]

A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction–diffusion systems on fixed and growing domains, J. Comput. Phys., 214 (2006), 239-263.  doi: 10.1016/j.jcp.2005.09.012.

[20]

A. Madzvamuse and A. H. Chung, Analysis and simulations of coupled bulk-surface reaction-diffusion systems on exponentially evolving volumes, Math. Model. Nat. Phenom., 11 (2016), 4-32.  doi: 10.1051/mmnp/201611502.

[21]

A. MadzvamuseE. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains, J. Math. Biol, 61 (2010), 133-164.  doi: 10.1007/s00285-009-0293-4.

[22]

A. Madzvamuse and P. K. Maini, Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains, J. Comput. Phys., 225 (2007), 100-119.  doi: 10.1016/j.jcp.2006.11.022.

[23]

J. Morgan, Global existence for semilinear parabolic systems, SIAM J. Math. Anal, 20 (1989), 1128-1144.  doi: 10.1137/0520075.

[24]

J. Morgan and V. Sharma, Global existence of solutions to volume-surface reaction diffusion systems with dynamic boundary conditions, Differential Integral Equations, 33 (2020), 113-139. 

[25]

R. G. PlazaF. Sànchez-GarduñoP. PadillaR. A. Barrio and P. K. Maini, The effect of growth and curvature on pattern formation, J. Dynam. Differential Equations, 16 (2004), 1093-1121.  doi: 10.1007/s10884-004-7834-8.

[26]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93-106.  doi: 10.1137/S0036144599359735.

[27]

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.

[28]

A. Rätz and M. Röger, Turing instabilities in a mathematical model for signaling networks, J. Math. Biol., 65 (2012), 1215-1244.  doi: 10.1007/s00285-011-0495-4.

[29]

A. Rätz and M. Röger, Symmetry breaking in a bulk-surface reaction-diffusion model for signaling networks, Nonlinearity, 27 (2014), 1805-1827. 

[30]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, Vol. 1072, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.

[31]

V. Sharma, Global existence and uniform estimates for solutions to reaction-diffusion systems with mass transport type of boundary conditions, Comunication on Pure and Applied Analysis, (2020)

[32]

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

[33]

V. Sharma and J. Morgan, Uniform bounds for solutions to volume-surface reaction diffusion systems, Differential Integral Equations, 30 (2017), 423-442. 

[34]

V. Sharma and J. V. Prajapat, Global existence of solution to volume surface reaction diffusion system with evolving domain, work-in-progress.

[35]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B. Biol. Sci., 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[36]

C. VenkataramanO. Lakkis and A. Madzvamuse, Global existence for semilinear reaction–diffusion systems on evolving domains, Journal of Mathematical Biology, 64 (2012), 41-67.  doi: 10.1007/s00285-011-0404-x.

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