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March  2021, 20(3): 955-974. doi: 10.3934/cpaa.2021001

Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions

Vandana Sharma

Department of Mathematics, Indian Institute of Technology Jodhpur, NH 65, Nagaur Road, Karwar, Jodhpur 342037, India

Received  May 2020 Revised  November 2020 Published  March 2021 Early access  January 2021

Fund Project: I acknowledge IIT Jodhpur for research grant support as SEED grant and infrastructural support

We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions. Under reasonable hypotheses, we establish the existence of component wise non-negative global solutions which are uniformly bounded in the sup norm.

Keywords: Reaction-diffusion equations, mass transport, conservation of mass, global wellposedness, linear estimates.
Mathematics Subject Classification: Primary: 35K57, 35B45.
Citation: Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001
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: Math. Theor., 40 (2007), 12335-12350.  doi: 10.1088/1751-8113/40/41/005.  Google Scholar

[2]

José A. Cãnizo, Laurent Desvillettes and Klemens Fellner, Improved duality estimates and applications to reaction-diffusion equations, Commun. Partial Differ. Equ., 39 (2014), 1185-1204.  doi: 10.1080/03605302.2013.829500.  Google Scholar

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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.  Google Scholar

[4]

Klemens Fellner, J. Morgan and Bao Quoc Tang, Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions, arXiv: 1906.06902. Google Scholar

[5]

Klemens Fellner, J. Morgan and Bao Quoc Tang, Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Annales de l'Institut Henri Poincaré, 37 (2020), 281-307.  doi: 10.1016/j.anihpc.2019.09.003.  Google Scholar

[6]

T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Éc. Norm. Supér., (4) (2010), 117–142. doi: 10.24033/asens.2117.  Google Scholar

[7]

Selwyn L. Hollis, Robert H. Jr. Martin and Michel Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744-761.  doi: 10.1137/0518057.  Google Scholar

[8] O. A. Ladyzhenskaia and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.   Google Scholar
[9]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I, 1968.  Google Scholar

[10]

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

[11]

J. Morgan and Bao Quoc Tang, Boundedness for reaction-diffusion systems with Lyapunov functions and intermediate sum conditions, Nonlinearity, 33 (2020), 3105-3133.  doi: 10.1088/1361-6544/ab8772.  Google Scholar

[12]

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

[13]

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

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

Bao Quoc Tang, Global classical solutions to reaction-diffusion systems in one and two dimensions, Commun. Math. Sci., 16 (2018), 411-423.  doi: 10.4310/CMS.2018.v16.n2.a5.  Google Scholar

[18]

M. E. Taylor, Partial Differential Equations I-III, Springer, 2011. doi: 10.1007/978-1-4419-7049-7.  Google Scholar

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: Math. Theor., 40 (2007), 12335-12350.  doi: 10.1088/1751-8113/40/41/005.  Google Scholar

[2]

José A. Cãnizo, Laurent Desvillettes and Klemens Fellner, Improved duality estimates and applications to reaction-diffusion equations, Commun. Partial Differ. Equ., 39 (2014), 1185-1204.  doi: 10.1080/03605302.2013.829500.  Google Scholar

[3]

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.  Google Scholar

[4]

Klemens Fellner, J. Morgan and Bao Quoc Tang, Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions, arXiv: 1906.06902. Google Scholar

[5]

Klemens Fellner, J. Morgan and Bao Quoc Tang, Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Annales de l'Institut Henri Poincaré, 37 (2020), 281-307.  doi: 10.1016/j.anihpc.2019.09.003.  Google Scholar

[6]

T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Éc. Norm. Supér., (4) (2010), 117–142. doi: 10.24033/asens.2117.  Google Scholar

[7]

Selwyn L. Hollis, Robert H. Jr. Martin and Michel Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744-761.  doi: 10.1137/0518057.  Google Scholar

[8] O. A. Ladyzhenskaia and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.   Google Scholar
[9]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R.I, 1968.  Google Scholar

[10]

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

[11]

J. Morgan and Bao Quoc Tang, Boundedness for reaction-diffusion systems with Lyapunov functions and intermediate sum conditions, Nonlinearity, 33 (2020), 3105-3133.  doi: 10.1088/1361-6544/ab8772.  Google Scholar

[12]

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

[13]

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

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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

[17]

Bao Quoc Tang, Global classical solutions to reaction-diffusion systems in one and two dimensions, Commun. Math. Sci., 16 (2018), 411-423.  doi: 10.4310/CMS.2018.v16.n2.a5.  Google Scholar

[18]

M. E. Taylor, Partial Differential Equations I-III, Springer, 2011. doi: 10.1007/978-1-4419-7049-7.  Google Scholar

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