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

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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

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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	O. A. Ladyzhenskaia and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
[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.
[10]	J. Morgan, Global existence for semilinear parabolic systems, SIAM J. Math. Anal., 20 (1989), 1128-1144. doi: 10.1137/0520075.
[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.
[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.
[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.
[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.
[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.
[16]	V. Sharma and J. Morgan, Uniform bounds for solutions to volume-surface reaction diffusion systems, Differ. Integral Equ., 30 (2017), 423-442.
[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.
[18]	M. E. Taylor, Partial Differential Equations I-III, Springer, 2011. doi: 10.1007/978-1-4419-7049-7.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[8]	O. A. Ladyzhenskaia and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
[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.
[10]	J. Morgan, Global existence for semilinear parabolic systems, SIAM J. Math. Anal., 20 (1989), 1128-1144. doi: 10.1137/0520075.
[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.
[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.
[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.
[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.
[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.
[16]	V. Sharma and J. Morgan, Uniform bounds for solutions to volume-surface reaction diffusion systems, Differ. Integral Equ., 30 (2017), 423-442.
[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.
[18]	M. E. Taylor, Partial Differential Equations I-III, Springer, 2011. doi: 10.1007/978-1-4419-7049-7.

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