December  2016, 21(10): 3441-3462. doi: 10.3934/dcdsb.2016106

Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions

1. 

Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstraße 36, 8010 Graz, Austria, Austria

2. 

Graduate School of Engineering Science, Department of Systems Innovation, Division of Mathematical Science, Osaka University, Japan

Received  January 2016 Revised  March 2016 Published  November 2016

This paper considers quadratic and super-quadratic reaction-diffusion systems, for which all species satisfy uniform-in-time $L^1$ a-priori estimates, for instance, as a consequence of suitable mass conservation laws. A new result on the global existence of classical solutions is proved in three and higher space dimensions by combining regularity and interpolation arguments in Bochner spaces, a bootstrap scheme and a weak comparison argument. Moreover, provided that the considered system allows for entropy entropy-dissipation estimates proving exponential convergence to equilibrium, we are also able to prove that solutions are bounded uniformly-in-time.
Citation: 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
References:
[1]

H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47.  Google Scholar

[2]

M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 125 (2006), 249-280. doi: 10.1007/s10955-005-8075-x.  Google Scholar

[3]

M. Bisi, F. Conforto and L. Desvillettes, Quasi-steady-state approximation for reaction-diffusion equations, Bull. Inst. Math., Acad. Sin. (N.S.), 2 (2007), 823-850.  Google Scholar

[4]

D. Bothe and M. Pierre, Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate, J. Math. Anal. Appl., 368 (2010), 120-132. doi: 10.1016/j.jmaa.2010.02.044.  Google Scholar

[5]

J. A. Cañizo, L. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Comm. Partial Differential Equations, 39 (2014), 1185-1204. doi: 10.1080/03605302.2013.829500.  Google Scholar

[6]

C. Caputo and A. Vasseur, Global regularity of solutions to systems of reaction-diffusion with sub-quadratic growth in any dimension, Commun. Partial Differential Equations, 34 (2009), 1228-1250. doi: 10.1080/03605300903089867.  Google Scholar

[7]

M. Chipot, Elements of Nonlinear Analysis, Birkhäuser, 2000. doi: 10.1007/978-3-0348-8428-0.  Google Scholar

[8]

A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamic Limits, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0086457.  Google Scholar

[9]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176. doi: 10.1016/j.jmaa.2005.07.003.  Google Scholar

[10]

L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a priori bounds, Revista Matemática Iberoamericana, 24 (2008), 407-431. doi: 10.4171/RMI/541.  Google Scholar

[11]

L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104. doi: 10.1007/978-3-662-45504-3_9.  Google Scholar

[12]

L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, About global existence for quadratic systems of reaction-diffusion, J. Advanced Nonlinear Studies, 7 (2007), 491-511. doi: 10.1515/ans-2007-0309.  Google Scholar

[13]

T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Ec. Norm. Super., (4) 43 (2010), 117-142.  Google Scholar

[14]

S. L. Hollis and J. J. Morgan, On the blow-up of solution to some semilinear and quasilinear reaction-diffusion systems, Rocky Mountain Journal of Mathematics, 24 (1994), 1447-1465. doi: 10.1216/rmjm/1181072348.  Google Scholar

[15]

G. Karali and T. Suzuki, Global-in-time behavior of the solution to a Gierer-Meinhardt system, Discrete and Continuous Dynamical Systems, 33 (2013), 2885-2900. doi: 10.3934/dcds.2013.33.2885.  Google Scholar

[16]

E. Latos, T. Suzuki and Y. Yamada, Transient and asymptotic dynamics of a prey-predator system with diffusion, Math. Meth. Appl. Sci., 35 (2012), 1101-1109. doi: 10.1002/mma.2524.  Google Scholar

[17]

M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168. doi: 10.1007/s000280300007.  Google Scholar

[18]

M. Pierre, Global existence in reaction-diffusion systems with dissipation of mass: A Survey, Milan J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4.  Google Scholar

[19]

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

[20]

F. Rothe, Global Solutions of Reaction-Diffusion Equations, Lecture Notes in Mathematics, Springer-Verlag, 1984.  Google Scholar

[21]

T. Suzuki and Y. Yamada, Global-in-time behavior of Lotka-Volterra system with diffusion, Indiana Univ. Math., 64 (2015), 181-216. doi: 10.1512/iumj.2015.64.5460.  Google Scholar

show all references

References:
[1]

H. Amann, Global existence for semilinear parabolic systems, J. Reine Angew. Math., 360 (1985), 47-83. doi: 10.1515/crll.1985.360.47.  Google Scholar

[2]

M. Bisi and L. Desvillettes, From reactive Boltzmann equations to reaction-diffusion systems, J. Stat. Phys., 125 (2006), 249-280. doi: 10.1007/s10955-005-8075-x.  Google Scholar

[3]

M. Bisi, F. Conforto and L. Desvillettes, Quasi-steady-state approximation for reaction-diffusion equations, Bull. Inst. Math., Acad. Sin. (N.S.), 2 (2007), 823-850.  Google Scholar

[4]

D. Bothe and M. Pierre, Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate, J. Math. Anal. Appl., 368 (2010), 120-132. doi: 10.1016/j.jmaa.2010.02.044.  Google Scholar

[5]

J. A. Cañizo, L. Desvillettes and K. Fellner, Improved duality estimates and applications to reaction-diffusion equations, Comm. Partial Differential Equations, 39 (2014), 1185-1204. doi: 10.1080/03605302.2013.829500.  Google Scholar

[6]

C. Caputo and A. Vasseur, Global regularity of solutions to systems of reaction-diffusion with sub-quadratic growth in any dimension, Commun. Partial Differential Equations, 34 (2009), 1228-1250. doi: 10.1080/03605300903089867.  Google Scholar

[7]

M. Chipot, Elements of Nonlinear Analysis, Birkhäuser, 2000. doi: 10.1007/978-3-0348-8428-0.  Google Scholar

[8]

A. De Masi and E. Presutti, Mathematical Methods for Hydrodynamic Limits, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0086457.  Google Scholar

[9]

L. Desvillettes and K. Fellner, Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations, J. Math. Anal. Appl., 319 (2006), 157-176. doi: 10.1016/j.jmaa.2005.07.003.  Google Scholar

[10]

L. Desvillettes and K. Fellner, Entropy methods for reaction-diffusion equations: Slowly growing a priori bounds, Revista Matemática Iberoamericana, 24 (2008), 407-431. doi: 10.4171/RMI/541.  Google Scholar

[11]

L. Desvillettes and K. Fellner, Exponential convergence to equilibrium for a nonlinear reaction-diffusion systems arising in reversible chemistry, System Modelling and Optimization, IFIP AICT, 443 (2014), 96-104. doi: 10.1007/978-3-662-45504-3_9.  Google Scholar

[12]

L. Desvillettes, K. Fellner, M. Pierre and J. Vovelle, About global existence for quadratic systems of reaction-diffusion, J. Advanced Nonlinear Studies, 7 (2007), 491-511. doi: 10.1515/ans-2007-0309.  Google Scholar

[13]

T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Ec. Norm. Super., (4) 43 (2010), 117-142.  Google Scholar

[14]

S. L. Hollis and J. J. Morgan, On the blow-up of solution to some semilinear and quasilinear reaction-diffusion systems, Rocky Mountain Journal of Mathematics, 24 (1994), 1447-1465. doi: 10.1216/rmjm/1181072348.  Google Scholar

[15]

G. Karali and T. Suzuki, Global-in-time behavior of the solution to a Gierer-Meinhardt system, Discrete and Continuous Dynamical Systems, 33 (2013), 2885-2900. doi: 10.3934/dcds.2013.33.2885.  Google Scholar

[16]

E. Latos, T. Suzuki and Y. Yamada, Transient and asymptotic dynamics of a prey-predator system with diffusion, Math. Meth. Appl. Sci., 35 (2012), 1101-1109. doi: 10.1002/mma.2524.  Google Scholar

[17]

M. Pierre, Weak solutions and supersolutions in $L^1$ for reaction-diffusion systems, J. Evol. Equ., 3 (2003), 153-168. doi: 10.1007/s000280300007.  Google Scholar

[18]

M. Pierre, Global existence in reaction-diffusion systems with dissipation of mass: A Survey, Milan J. Math., 78 (2010), 417-455. doi: 10.1007/s00032-010-0133-4.  Google Scholar

[19]

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

[20]

F. Rothe, Global Solutions of Reaction-Diffusion Equations, Lecture Notes in Mathematics, Springer-Verlag, 1984.  Google Scholar

[21]

T. Suzuki and Y. Yamada, Global-in-time behavior of Lotka-Volterra system with diffusion, Indiana Univ. Math., 64 (2015), 181-216. doi: 10.1512/iumj.2015.64.5460.  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]

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, 2021, 14 (2) : 389-406. doi: 10.3934/krm.2021009

[3]

Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043

[4]

Prasanta Kumar Barik. Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation. Evolution Equations & Control Theory, 2020, 9 (2) : 431-446. doi: 10.3934/eect.2020012

[5]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891

[6]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65

[7]

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

[8]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39

[9]

Gheorghe Craciun, Jiaxin Jin, Casian Pantea, Adrian Tudorascu. Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1305-1335. doi: 10.3934/dcdsb.2020164

[10]

Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042

[11]

Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304

[12]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519

[13]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242

[14]

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

[15]

Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025

[16]

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

[17]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[18]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815

[19]

José A. Carrillo, Renjun Duan, Ayman Moussa. Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. Kinetic & Related Models, 2011, 4 (1) : 227-258. doi: 10.3934/krm.2011.4.227

[20]

Siqing Li, Zhonghua Qiao. A meshless collocation method with a global refinement strategy for reaction-diffusion systems on evolving domains. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021057

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (13)

[Back to Top]