American Institute of Mathematical Sciences

Advanced
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  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 & Continuous Dynamical Systems - S, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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 & Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106
[2]

Shaohua Chen, Runzhang Xu, Hongtao Yang. Global and blowup solutions for general Lotka-Volterra systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1757-1768. doi: 10.3934/cpaa.2016012
[3]

Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55
[4]

Zhi-Cheng Wang, Hui-Ling Niu, Shigui Ruan. On the existence of axisymmetric traveling fronts in Lotka-Volterra competition-diffusion systems in ℝ3. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1111-1144. doi: 10.3934/dcdsb.2017055
[5]

Norimichi Hirano, Wieslaw Krawcewicz, Haibo Ruan. Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 709-735. doi: 10.3934/dcds.2011.30.709
[6]

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
[7]

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
[8]

Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135
[9]

Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069
[10]

Yuan Lou, Salomé Martínez, Wei-Ming Ni. On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 175-190. doi: 10.3934/dcds.2000.6.175
[11]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[12]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111
[13]

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
[14]

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 & Continuous Dynamical Systems - A, 1995, 1 (1) : 103-117. doi: 10.3934/dcds.1995.1.103
[15]

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 & Applied Analysis, 2020, 19 (1) : 1-18. doi: 10.3934/cpaa.2020001
[16]

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
[17]

Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[18]

Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 137-144. doi: 10.3934/mbe.2006.3.137
[19]

Guichen Lu, Zhengyi Lu. Permanence for two-species Lotka-Volterra cooperative systems with delays. Mathematical Biosciences & Engineering, 2008, 5 (3) : 477-484. doi: 10.3934/mbe.2008.5.477
[20]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043

2018 Impact Factor: 0.545

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences