-
Previous Article
Approximation of a stochastic two-phase flow model by a splitting-up method
- CPAA Home
- This Issue
-
Next Article
The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law
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 |
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.
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. 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. |
[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. 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. |
[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. |
[1] |
Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021005 |
[2] |
Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2021004 |
[3] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
[4] |
Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223 |
[5] |
Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427 |
[6] |
Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024 |
[7] |
Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311 |
[8] |
Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 |
[9] |
Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021009 |
[10] |
Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021006 |
[11] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[12] |
Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 |
[13] |
Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021014 |
[14] |
Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617 |
[15] |
Luigi Barletti, Giovanni Nastasi, Claudia Negulescu, Vittorio Romano. Mathematical modelling of charge transport in graphene heterojunctions. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021010 |
[16] |
Jingni Guo, Junxiang Xu, Zhenggang He, Wei Liao. Research on cascading failure modes and attack strategies of multimodal transport network. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2020159 |
[17] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[18] |
Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 |
[19] |
Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 |
[20] |
Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 |
2019 Impact Factor: 1.105
Tools
Metrics
Other articles
by authors
[Back to Top]