American Institute of Mathematical Sciences

Advanced
February  2021, 14(2): 635-651. doi: 10.3934/dcdss.2020334

Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions

Klemens Fellner 1,, , Jeff Morgan 2, and  Bao Quoc Tang 3,

1. 

Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
2. 

Department of Mathematics, University of Houston, Houston, Texas 77004, USA
3. 

Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria

* Corresponding author: Klemens Fellner

Received  June 2019 Published  February 2021 Early access  April 2020

Fund Project: This work is supported by the International Training Program IGDK 1754 and NAWI Graz

Uniform-in-time bounds of nonnegative classical solutions to reaction-diffusion systems in all space dimension are proved. The systems are assumed to dissipate the total mass and to have locally Lipschitz nonlinearities of at most (slightly super-) quadratic growth. This pushes forward the recent advances concerning global existence of reaction-diffusion systems dissipating mass in which a uniform-in-time bound has been known only in space dimension one or two. As an application, skew-symmetric Lotka-Volterra systems are shown to have unique classical solutions which are uniformly bounded in time in all dimensions with relatively compact trajectories in $ C(\overline{\Omega})^m $.

Keywords: Reaction-diffusion systems, Mass dissipation, Classical solutions, Global existence, Uniform boundedness, Lotka-Volterra systems.
Mathematics Subject Classification: 35A01, 35K57, 35K58, 35Q92, 92D25.
Citation: Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334
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.

[2]

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

[3]

M. C. Caputo, T. Goudon and A. Vasseur, Solutions of the 4-species quadratic reaction-diffusion system are bounded and $C^\infty$-smooth, in any space dimension, Analysis and PDEs, 12 (2019), 1773-1804.  doi: 10.2140/apde.2019.12.1773.

[4]

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

[5]

E. Conway, D. Hoff and J. Smoller, Large time behavior of nonlinear reaction diffusion systems, SIAM J. Appl. Math., 35 (1978), 1-16. 

[6]

B. P. Cupps, J. Morgan and B. Q. Tang, Uniform boundedness for reaction-diffusion systems with mass dissipation, arXiv: 1905.10599.

[7]

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

[8]

K. Fellner, J. Morgan and B. Q. Tang, Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Annales de l'Institut H. Poincaré C, Analyse Non Linéaire, 37 (2020), 281-307.  doi: 10.1016/j.anihpc.2019.09.003.

[9]

K. Fellner and B. Q. Tang, Explicit exponential convergence to equilibrium for nonlinear reaction-diffusion systems with detailed balance condition, Nonlinear Analysis, 159 (2017), 145-180.  doi: 10.1016/j.na.2017.02.007.

[10]

K. Fellner and B. Q. Tang, Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems, Zeitschrift für Angewandte Mathematik und Physik, 69 (2018), Art. 54, 30 pp. doi: 10.1007/s00033-018-0948-3.

[11]

J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Rational Mech. Anal., 218 (2015), 553-587.  doi: 10.1007/s00205-015-0866-x.

[12]

J. Fischer, Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations, Nonlinear Anal., 159 (2017), 181-207.  doi: 10.1016/j.na.2017.03.001.

[13]

W. B. Fitzgibbon, S. L. Hollis and J. J. Morgan, Stability and Lyapunov functions for reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 28 (1997), 595-610.  doi: 10.1137/S0036141094272241.

[14]

T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Annales Scientifiques de L'École Normale Supérieure, 43 (2010), 117-142.  doi: 10.24033/asens.2117.

[15]

S. L. Hollis, R. H. Martin, Jr. and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 18 (1987), 744-761.  doi: 10.1137/0518057.

[16]

Ya. I. Kanel', Solvability in the large of a system of reaction-diffusion equations with the balance condition, Differentsialýe Uravneniya, 26 (1990), 448-458. 

[17]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I., 1968.

[18]

D. Lamberton, Equations d'évolution linéaires associées à des semi-groupes de contraction dans les espaces $L^p$, J. Functional Anal., 72 (1987), 252-262.  doi: 10.1016/0022-1236(87)90088-7.

[19]

J. Morgan, Global existence for semilinear parabolic systems, SIAM Journal on Mathematical Analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.

[20]

J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 21 (1990), 1172-1189.  doi: 10.1137/0521064.

[21]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan Journal of Mathematics, 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.

[22]

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

[23]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Journal on Mathematical Analysis, 28 (1997), 259-269.  doi: 10.1137/S0036141095295437.

[24]

M. Pierre, T. Suzuki and Y. Yamada, Dissipative reaction-diffusion systems with quadratic growth, Indiana University Mathematics Journal, 68 (2019), 291-322.  doi: 10.1512/iumj.2019.68.7447.

[25]

F. Rothe, Global Solutions of Reaction-diffusion Systems, Lecture Notes in Mathematics, 1072. Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.

[26]

P. Souplet, Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth, Journal of Evolution Equations, 18 (2018), 1713-1720.  doi: 10.1007/s00028-018-0458-y.

[27]

T. Suzuki and Y. Yamada, A Lotka-Volterra system with diffusion, Nonlinear Analysis in Interdisciplinary Sciences-Modellings, Theory and Simulations, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkotosho, Tokyo, 36 (2013), 215-236. 

[28]

B. Q. Tang, Global classical solutions to reaction-diffusion systems in one and two dimensions, Communications in Mathematical Sciences, 16 (2018), 411-423.  doi: 10.4310/CMS.2018.v16.n2.a5.

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.

[2]

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

[3]

M. C. Caputo, T. Goudon and A. Vasseur, Solutions of the 4-species quadratic reaction-diffusion system are bounded and $C^\infty$-smooth, in any space dimension, Analysis and PDEs, 12 (2019), 1773-1804.  doi: 10.2140/apde.2019.12.1773.

[4]

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

[5]

E. Conway, D. Hoff and J. Smoller, Large time behavior of nonlinear reaction diffusion systems, SIAM J. Appl. Math., 35 (1978), 1-16. 

[6]

B. P. Cupps, J. Morgan and B. Q. Tang, Uniform boundedness for reaction-diffusion systems with mass dissipation, arXiv: 1905.10599.

[7]

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

[8]

K. Fellner, J. Morgan and B. Q. Tang, Global classical solutions to quadratic systems with mass control in arbitrary dimensions, Annales de l'Institut H. Poincaré C, Analyse Non Linéaire, 37 (2020), 281-307.  doi: 10.1016/j.anihpc.2019.09.003.

[9]

K. Fellner and B. Q. Tang, Explicit exponential convergence to equilibrium for nonlinear reaction-diffusion systems with detailed balance condition, Nonlinear Analysis, 159 (2017), 145-180.  doi: 10.1016/j.na.2017.02.007.

[10]

K. Fellner and B. Q. Tang, Convergence to equilibrium of renormalised solutions to nonlinear chemical reaction-diffusion systems, Zeitschrift für Angewandte Mathematik und Physik, 69 (2018), Art. 54, 30 pp. doi: 10.1007/s00033-018-0948-3.

[11]

J. Fischer, Global existence of renormalized solutions to entropy-dissipating reaction-diffusion systems, Arch. Rational Mech. Anal., 218 (2015), 553-587.  doi: 10.1007/s00205-015-0866-x.

[12]

J. Fischer, Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations, Nonlinear Anal., 159 (2017), 181-207.  doi: 10.1016/j.na.2017.03.001.

[13]

W. B. Fitzgibbon, S. L. Hollis and J. J. Morgan, Stability and Lyapunov functions for reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 28 (1997), 595-610.  doi: 10.1137/S0036141094272241.

[14]

T. Goudon and A. Vasseur, Regularity analysis for systems of reaction-diffusion equations, Annales Scientifiques de L'École Normale Supérieure, 43 (2010), 117-142.  doi: 10.24033/asens.2117.

[15]

S. L. Hollis, R. H. Martin, Jr. and M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 18 (1987), 744-761.  doi: 10.1137/0518057.

[16]

Ya. I. Kanel', Solvability in the large of a system of reaction-diffusion equations with the balance condition, Differentsialýe Uravneniya, 26 (1990), 448-458. 

[17]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I., 1968.

[18]

D. Lamberton, Equations d'évolution linéaires associées à des semi-groupes de contraction dans les espaces $L^p$, J. Functional Anal., 72 (1987), 252-262.  doi: 10.1016/0022-1236(87)90088-7.

[19]

J. Morgan, Global existence for semilinear parabolic systems, SIAM Journal on Mathematical Analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.

[20]

J. Morgan, Boundedness and decay results for reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 21 (1990), 1172-1189.  doi: 10.1137/0521064.

[21]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan Journal of Mathematics, 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.

[22]

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

[23]

M. Pierre and D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Journal on Mathematical Analysis, 28 (1997), 259-269.  doi: 10.1137/S0036141095295437.

[24]

M. Pierre, T. Suzuki and Y. Yamada, Dissipative reaction-diffusion systems with quadratic growth, Indiana University Mathematics Journal, 68 (2019), 291-322.  doi: 10.1512/iumj.2019.68.7447.

[25]

F. Rothe, Global Solutions of Reaction-diffusion Systems, Lecture Notes in Mathematics, 1072. Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0099278.

[26]

P. Souplet, Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth, Journal of Evolution Equations, 18 (2018), 1713-1720.  doi: 10.1007/s00028-018-0458-y.

[27]

T. Suzuki and Y. Yamada, A Lotka-Volterra system with diffusion, Nonlinear Analysis in Interdisciplinary Sciences-Modellings, Theory and Simulations, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkotosho, Tokyo, 36 (2013), 215-236. 

[28]

B. Q. Tang, Global classical solutions to reaction-diffusion systems in one and two dimensions, Communications in Mathematical Sciences, 16 (2018), 411-423.  doi: 10.4310/CMS.2018.v16.n2.a5.

[1]

Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
[2]

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
[3]

Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012
[4]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55
[5]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055
[6]

Norimichi Hirano, Wieslaw Krawcewicz, Haibo Ruan. Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 709-735. doi: 10.3934/dcds.2011.30.709
[7]

Vandana Sharma, Jyotshana V. Prajapat. Global existence of solutions to reaction diffusion systems with mass transport type boundary conditions on an evolving domain. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 109-135. doi: 10.3934/dcds.2021109
[8]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815
[9]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891
[10]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5141-5164. doi: 10.3934/dcds.2021071
[11]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[12]

Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069
[13]

Yuan Lou, Salomé Martínez, Wei-Ming Ni. On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 175-190. doi: 10.3934/dcds.2000.6.175
[14]

Chiun-Chuan Chen, Yin-Liang Huang, Li-Chang Hung, Chang-Hong Wu. Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats. Communications on Pure and Applied Analysis, 2020, 19 (1) : 1-18. doi: 10.3934/cpaa.2020001
[15]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[16]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111
[17]

C. M. Evans, G. L. Findley. Analytic solutions to a class of two-dimensional Lotka-Volterra dynamical systems. Conference Publications, 2001, 2001 (Special) : 137-142. doi: 10.3934/proc.2001.2001.137
[18]

Tongren Ding, Hai Huang, Fabio Zanolin. A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 103-117. doi: 10.3934/dcds.1995.1.103
[19]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[20]

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

2021 Impact Factor: 1.865

Tools

Metrics

  • PDF downloads (256)
  • HTML views (346)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy