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

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 $.

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, 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.  Google Scholar

[2]

J. A. CañizoL. 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. CaputoT. 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

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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. ConwayD. 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. DesvillettesK. FellnerM. 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. FellnerJ. 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. FitzgibbonS. 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. HollisR. H. MartinJr. 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. PierreT. 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. CañizoL. 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. CaputoT. 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. ConwayD. 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. DesvillettesK. FellnerM. 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. FellnerJ. 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. FitzgibbonS. 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. HollisR. H. MartinJr. 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. PierreT. 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

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