Compétition Réaction-Diffusion et comportement asymptotique d’un problème d’obstacle doublement non linéaire
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 2, p. 345-362

In this paper, we study the competition between the diffusion and the reaction for the problem of type $\beta {\left(w\right)}_{t}-{d}_{\epsilon }\phantom{\rule{0.222222em}{0ex}}diva\left(x,\phantom{\rule{0.222222em}{0ex}}Dw\right)+{r}_{\epsilon }\phantom{\rule{0.222222em}{0ex}}g\left(x,\beta \left(w\right)\right)=f,$ where $\mathbf{a}$ is a Lerray-Lions operator, $\beta$ is a nondecreasing continuous function and the reaction $g$ is a nondecreasing function that depend on the space $x$. Assume that, the coefficient of diffusion ${d}_{\epsilon }$ and the reaction ${r}_{\epsilon }$ depend on the parameter $\epsilon$ with ${d}_{\epsilon }$ and/or ${r}_{\epsilon }$ tends to $+\infty$ as $\epsilon \to 0$. In the case when, the reaction coefficient is very fast, we study the asymptotic behavior as $t\to \infty$ of the solution of the obstacle problem to characterize the initial data for the limit problem.

Le but de cet article est l’étude de la compétition Réaction-Diffusion pour un problème de type $\beta {\left(w\right)}_{t}-{d}_{\epsilon }\phantom{\rule{0.222222em}{0ex}}diva\left(x,\phantom{\rule{0.222222em}{0ex}}Dw\right)+{r}_{\epsilon }\phantom{\rule{0.222222em}{0ex}}g\left(x,\beta \left(w\right)\right)=f,$$\mathbf{a}$ est un opérateur de Lerray-Lions, $\beta$ est une fonction continue croissante et la réaction $g$ est une fonction croissante qui dépend de l’espace $x$. On suppose que les coefficients de diffusion ${d}_{\epsilon }$ et de Réaction ${r}_{\epsilon }$ dépendent du paramètre $\epsilon$ avec ${d}_{\epsilon }$ et/ou ${r}_{\epsilon }$ tends vers $+\infty$ lorsque $\epsilon \to 0$. Dans le cas où, le coefficient de réaction est très rapide, nous étudions le comportement asymptotique lorsque $t\to \infty$ de la solution du problème d’obstacle afin de caractériser la donnée initiale du problème limite.

Accepted : 2009-06-26
Published online : 2010-09-01
DOI : https://doi.org/10.5802/afst.1246
@article{AFST_2010_6_19_2_345_0,
author = {Fahd Karami},
title = {Comp\'etition R\'eaction-Diffusion et comportement asymptotique d'un probl\`eme d'obstacle doublement non lin\'eaire},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {6e s{\'e}rie, 19},
number = {2},
year = {2010},
pages = {345-362},
doi = {10.5802/afst.1246},
mrnumber = {2674766},
zbl = {1203.35147},
language = {fr},
url = {https://afst.centre-mersenne.org/item/AFST_2010_6_19_2_345_0}
}
Karami, Fahd. Compétition Réaction-Diffusion et comportement asymptotique d’un problème d’obstacle doublement non linéaire. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 2, pp. 345-362. doi : 10.5802/afst.1246. afst.centre-mersenne.org/item/AFST_2010_6_19_2_345_0/

[1] F. Andreu, J. M. Mazon, J, Toledo, Asymptotic behaviour of solutions of quasi-linear parabolic equations with non linear flux, Comput. Appl. Math. 17 (1998) 201-215. | MR 1690131 | Zbl 0914.35058

[2] J. Arrieta, A. N. Carvalho and A. Rodrigues-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Diff. Equations, 168 (2000) 33-59. | MR 1801342 | Zbl 0963.35024

[3] Ph. Benilan, L. Boccardo, Th. Gallouet, R. Gariepy, M. Pierre and J .L. Vazquez, An $\phantom{\rule{0.166667em}{0ex}}{L}^{1}\phantom{\rule{0.166667em}{0ex}}$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22(2) (1995) 241-273. | Numdam | MR 1354907 | Zbl 0866.35037

[4] Ph. Benilan, Equations d’ évolution dans un espace de Banach quelconque et applications, Thesis,Univ. Orsay, 1972.

[5] Ph. Benilan , M. G. Crandall and A. Pazy, Evolution Equations Governed by Accretive Operators, Book to appear

[6] D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction, Adv. Differential Equations, 6, no. 10 (2001) 1173-1218.

[7] D. Bothe, The instantaneous limit of a reaction-diffusion system. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), 215–224, Lecture Notes in Pure and Appl. Math. 215, Dekker, New York, 2001. | MR 1818003 | Zbl 0973.35103

[8] H. Brezis, Opérateurs maximaux Monotones et semi-groupes de contractions dans les Espaces de Hilbert, Oxford Univ. Press, Oxford, 1984.

[9] H. Brezis and A. Pazy, Convergence And Approximation of Semigroupes of Nonlinear Operators in Banach Spaces, J . Func. Anal., 9 (1972) 63-74. | MR 293452 | Zbl 0231.47036

[10] E. C. M. Crooks, E. N. Dancer, D. Hilhorst, M. Mimura and H. Ninomiya,Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions, Nonlinear Anal. Real World Appl. 5, no. 4 (2004) 645-665. | MR 2079274 | Zbl 1079.35008

[11] L. C. Evans, M. Feldman and R. F. Gariepy, Fast/Slow diffusion and collapsing sandpiles, J. Differential Equations, 137 (1997) 166-209. | MR 1451539 | Zbl 0879.35019

[12] M. Henry, D. Hilhorst and Y. Nishiura, Singular limit of a second order nonlocal parabolic equation of conservative type arising in the micro-phase separation of diblock copolymers, Hokkaido Math. J. 32, no. 3 (2003) 561-622. | MR 2020592 | Zbl 1041.35008

[13] D. Hilhorst, M. Mimura and R. Weidenfeld, Singular limit of a class of non-cooperative reaction-diffusion systems, Taiwanese J. Math. 7 (2003), no. 3, 391–421. | MR 1998758 | Zbl 1153.35344

[14] J. K. Hale, Large diffusivity and asymptotic behavior in parabolic systems, J. Math. Anal. Appl. 1181(986) 455-466. | MR 852171 | Zbl 0602.35059

[15] D. Hilhorst, R. van der Hout and L. Peletier, A. Nonlinear diffusion in the presence of fast reaction, Nonlinear Anal. 41 (2000), no. 5-6, Ser. A : Theory Methods, 803–823. | MR 1780646 | Zbl 0963.35103

[16] J.L. Lions, Quelques méthodes de Résolution des problèmes aux Limites non Linéaires, Paris : Dunod (1969). | MR 259693 | Zbl 0189.40603

[17] N. Igbida, Stabilization for degenerate diffusion with absorption, Nonlinear Analysis 54 (2003) 93-107. | MR 1978967 | Zbl 1024.35055

[18] N. Igbida and F. Karami, Some competition Phenomena in Evolution Equations, Adv. Math. Sci. Appli. Vol. 17, No. 2 (2007) 559-587. | MR 2374141 | Zbl 1140.35342

[19] N. Igbida and F. Karami, Elliptic-Parabolic Equation with Absorption of Obstacle type, Soumis.

[20] N. Igbida, A nonlinear diffusion problem with localized large diffusion, Comm. Partial Differential Equations, 29, no. 5-6 (2004) 647-670. | MR 2059144 | Zbl 1065.35137

[21] J. M. Mazon and J. Toledo, Asymptotic behavior of solutions of the filtration equation in bounded domains, Dynam. Systems Appl, 3 (1994) 275-295. | MR 1272994 | Zbl 0797.35094

[22] M. Mimura, Reaction-diffusion systems arising in biological and chemical systems : application of singular limit procedures. Mathematical aspects of evolving interfaces(Funchal, 2000), 89–121, Lecture Notes in Math, 1812, Springer, Berlin, 2003. | MR 2011034 | Zbl 1030.35001

[23] A. Rodrigues-Bernal, Localized spatial homogenization and large diffusion, SIAM J. Math. Anal. 29 (1998)1361-1380. | MR 1638046 | Zbl 0915.35009