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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 |
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 $.
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. 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. |
[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. 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. |
[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. |
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