Global classical solutions to quadratic systems with mass control in arbitrary dimensions
Fellner, Klemens1 ; Morgan, Jeff2 ; Tang, Bao Quoc1
1 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
2 Department of Mathematics, University of Houston, Houston, TX 77204, USA
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 281-307.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

The global existence of classical solutions to reaction-diffusion systems in arbitrary space dimensions is studied. The nonlinearities are assumed to be quasi-positive, to have (slightly super-) quadratic growth, and to possess a mass control, which includes the important cases of mass conservation and mass dissipation. Under these assumptions, the local classical solution is shown to be global, and in the case of mass conservation or mass dissipation, to have the L∞-norm growing at most polynomially in time. Applications include skew-symmetric Lotka-Volterra systems and quadratic reversible chemical reactions.

DOI : 10.1016/j.anihpc.2019.09.003
Classification : 35A01, 35K57, 35K58, 35Q92
Mots-clés : Reaction-diffusion systems, Classical solutions, Global existence, Slowly-growing a-priori estimates, Mass dissipation
Fellner, Klemens 1 ; Morgan, Jeff 2 ; Tang, Bao Quoc 1

1 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
2 Department of Mathematics, University of Houston, Houston, TX 77204, USA 
@article{AIHPC_2020__37_2_281_0,
     author = {Fellner, Klemens and Morgan, Jeff and Tang, Bao Quoc},
     title = {Global classical solutions to quadratic systems with mass control in arbitrary dimensions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {281--307},
     publisher = {Elsevier},
     volume = {37},
     number = {2},
     year = {2020},
     doi = {10.1016/j.anihpc.2019.09.003},
     mrnumber = {4072804},
     zbl = {1433.35167},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.09.003/}
}
TY  - JOUR
AU  - Fellner, Klemens
AU  - Morgan, Jeff
AU  - Tang, Bao Quoc
TI  - Global classical solutions to quadratic systems with mass control in arbitrary dimensions
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2020
SP  - 281
EP  - 307
VL  - 37
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2019.09.003/
DO  - 10.1016/j.anihpc.2019.09.003
LA  - en
ID  - AIHPC_2020__37_2_281_0
ER  - 
%0 Journal Article
%A Fellner, Klemens
%A Morgan, Jeff
%A Tang, Bao Quoc
%T Global classical solutions to quadratic systems with mass control in arbitrary dimensions
%J Annales de l'I.H.P. Analyse non linéaire
%D 2020
%P 281-307
%V 37
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2019.09.003/
%R 10.1016/j.anihpc.2019.09.003
%G en
%F AIHPC_2020__37_2_281_0
Fellner, Klemens; Morgan, Jeff; Tang, Bao Quoc. Global classical solutions to quadratic systems with mass control in arbitrary dimensions. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 281-307. doi : 10.1016/j.anihpc.2019.09.003. http://www.numdam.org/articles/10.1016/j.anihpc.2019.09.003/

[1] Amann, H. Global existence for semilinear parabolic systems, J. Reine Angew. Math., Volume 360 (1985), pp. 47–83 | MR | Zbl

[2] Cáceres, M.J.; Cañizo, José A. Close-to-equilibrium behaviour of quadratic reaction-diffusion systems with detailed balance, Nonlinear Anal., Volume 159 (2017), pp. 62–84 | DOI | MR | Zbl

[3] Cañizo, J.A.; Desvillettes, L.; Fellner, K. Improved duality estimates and applications to reaction-diffusion equations, Commun. Partial Differ. Equ., Volume 39 (2014) no. 6, pp. 1185–1204 | DOI | MR | Zbl

[4] Caputo, M.C.; Goudon, T.; Vasseur, A.F. Solution of the 4-species quadratic reaction-diffusion system are bounded and C∞-smooth, in any space dimension, Anal. PDE, Volume 12 (2019) no. 7, pp. 1773–1804 | DOI | MR | Zbl

[5] Desvillettes, L.; Fellner, K. Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds, Rev. Mat. Iberoam., Volume 24 (2008) no. 2, pp. 407–431 | DOI | MR | Zbl

[6] Desvillettes, L.; Fellner, K.; Pierre, M.; Vovelle, J. About global existence for quadratic systems of reaction-diffusion, Adv. Nonlinear Stud., Volume 7 (2007) no. 3, pp. 491–511 | DOI | MR | Zbl

[7] Desvillettes, L.; Fellner, K.; Tang, B.Q. Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks, SIAM J. Math. Anal., Volume 49 (2017) no. 4, pp. 2666–2709 | DOI | MR | Zbl

[8] Dungey, N. Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z., Volume 247 (2004), pp. 765–794 | DOI | MR | Zbl

[9] Fellner, K.; Tang, B.Q. Explicit exponential convergence to equilibrium for nonlinear reaction-diffusion systems with detailed balance condition, Nonlinear Anal., Volume 159 (2017), pp. 145–180 | DOI | MR | Zbl

[10] Fellner, K.; Tang, B.Q. Convergence to equilibrium for renormalised solutions to nonlinear chemical reaction-diffusion systems, Z. Angew. Math. Phys., Volume 69 (2018) no. 3 (first online article 54) | DOI | MR | Zbl

[11] Fellner, K.; Latos, E.; Suzuki, T. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions, Discrete Contin. Dyn. Syst., Ser. B, Volume 21 (2016) no. 10, pp. 3441–3462 | MR | Zbl

[12] Fellner, K.; Latos, E.; Tang, B.Q. Global regularity and convergence to equilibrium of reaction-diffusion systems with nonlinear diffusion, J. Evol. Equ. (2019) (to appear in) | MR | Zbl

[13] Goudon, Th.; Vasseur, A. Regularity analysis for systems of reaction-diffusion equations, Ann. Sci. Ec. Norm. Super. (4), Volume 43 (2010) no. 1, pp. 117–142 | Numdam | MR | Zbl

[14] Henry, D. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981 | DOI | MR | Zbl

[15] Hollis, S.; Martin, R.H.; Pierre, M. Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., Volume 18 (1987), pp. 744–761 | DOI | MR | Zbl

[16] Kanel, J.I. Solvability in the large of a system of reaction-diffusion equations with the balance condition, Differ. Uravn., Volume 26 (1990), pp. 448–458 English translation: Differential Equations 26 (1990), 331–339 | MR | Zbl

[17] Krylov, N.V.; Safonov, M.V. A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 44 (1980) no. 1, pp. 161–175 | MR | Zbl

[18] Ladyženskaja, O.; Solonnikov, V.A.; Ural'ceva, N.N. Linear and Quasi-Linear Equations of Parabolic Type, Vol. 23, American Mathematical Soc., 1968 | DOI | Zbl

[19] Latos, E.; Suzuki, T.; Yamada, Y. Transient and asymptotic dynamics of a prey-predator system with diffusion, Math. Methods Appl. Sci., Volume 35 (2012) no. 9, pp. 1101–1109 | DOI | MR | Zbl

[20] Lieberman, G.M. Second Order Parabolic Differential Equations, World Scientific, 1996 | DOI | MR | Zbl

[21] Li, P.; Yau, S.T. On the parabolic kernel of the Schrödinger operator, Acta Math., Volume 156 (1986), pp. 153–201 | MR | Zbl

[22] Mora, X. Semilinear parabolic problems define semiflows on Ck spaces, Trans. Am. Math. Soc., Volume 278 (1983), pp. 21–55 | MR | Zbl

[23] Morgan, J. Global existence for seminlinear parabolic systems, SIAM J. Math. Anal., Volume 20 (1989), pp. 1128–1144 | DOI | MR | Zbl

[24] Nash, J. Continuity of solutions of parabolic and elliptic equations, Am. J. Math., Volume 80 (1958) no. 4, pp. 931–954 | DOI | MR | Zbl

[25] Nittka, R. Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains, J. Differ. Equ., Volume 251 (2011), pp. 860–880 | DOI | MR | Zbl

[26] Pierre, M. Global existence in reaction-diffusion systems with control of mass: a survey, Milan J. Math., Volume 78 (2010) no. 2, pp. 417–455 | DOI | MR | Zbl

[27] Pao, C.V. Nonlinear Parabolic and Elliptic Equations, Springer Science+Business Media, New York, 1992 | MR | Zbl

[28] Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer, 1983 | DOI | MR | Zbl

[29] Pierre, M. Weak solutions and super-solutions in L1 for reaction-diffusion systems, Nonlinear Evolution Equations and Related Topics, Birkhäuser, Basel, 2003, pp. 153–168 | MR | Zbl

[30] Pierre, M.; Schmitt, D. Blowup in reaction-diffusion systems with dissipation of mass, SIAM J. Math. Anal., Volume 28 (1997) no. 2, pp. 259–269 | DOI | MR | Zbl

[31] Pierre, M.; Suzuki, T.; Yamada, Y. Dissipative reaction-diffusion systems with quadratic growth, Indiana Univ. Math. J., Volume 68 (2019) no. 1, pp. 291–322 | DOI | MR | Zbl

[32] Pierre, M.; Suzuki, T.; Zou, R. Asymptotic behavior of solutions to chemical reaction-diffusion systems, J. Math. Anal. Appl., Volume 450 (2017) no. 1, pp. 152–168 | DOI | MR | Zbl

[33] Rothe, F. Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, vol. 1072, Springer-Verlag, Berlin, 1984 | DOI | MR | Zbl

[34] Souplet, P. Global existence for reaction–diffusion systems with dissipation of mass and quadratic growth, J. Evol. Equ., Volume 18 (2018), pp. 1713–1720 | DOI | MR | Zbl

[35] Suzuki, T.; Yamada, Y. A Lotka-Volterra system with diffusion, Nonlinear Analysis in Interdisciplinary Sciences-Modellings, Theory and Simulations, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 36, Gakkotosho, Tokyo, 2013, pp. 215–236 | MR

[36] Tang, Bao Q. Close-to-equilibrium regularity for reaction-diffusion systems, J. Evol. Equ., Volume 18 (2018) no. 2, pp. 845–869 | MR | Zbl

[37] Tang, Bao Q. Global classical solutions to reaction-diffusion systems in one and two dimensions, Commun. Math. Sci., Volume 16 (2018) no. 2, pp. 411–423 | MR | Zbl

[38] Wang, J. Global heat kernel estimates, Pac. J. Math., Volume 178 (1997) no. 2 | DOI | MR | Zbl

[39] Winkler, M. Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., Volume 248 (2010), pp. 2889–2905 | DOI | MR | Zbl

Cité par Sources :