American Institute of Mathematical Sciences

Advanced
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  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

[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

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

[1]

Shin-Ichiro Ei, Shyuh-Yaur Tzeng. Spike solutions for a mass conservation reaction-diffusion system. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3357-3374. doi: 10.3934/dcds.2020049
[2]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
[3]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
[4]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173
[5]

Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334
[6]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083
[7]

David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311
[8]

Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160
[9]

Jesus Garcia Azorero, Juan J. Manfredi, I. Peral, Julio D. Rossi. Limits for Monge-Kantorovich mass transport problems. Communications on Pure & Applied Analysis, 2008, 7 (4) : 853-865. doi: 10.3934/cpaa.2008.7.853
[10]

Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182
[11]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147
[12]

Aníbal Rodríguez-Bernal, Alejandro Vidal-López. A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (2) : 635-644. doi: 10.3934/cpaa.2014.13.635
[13]

Tarik Mohammed Touaoula. Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2473-2490. doi: 10.3934/cpaa.2020108
[14]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43
[15]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281
[16]

Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020087
[17]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005
[18]

Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47
[19]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129
[20]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (154)
  • HTML views (182)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences