Compétition Réaction-Diffusion et comportement asymptotique d’un problème d’obstacle doublement non linéaire

Karami, Fahd

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 2, pp. 345-362.

Le but de cet article est l’étude de la compétition Réaction-Diffusion pour un problème de type ${\displaystyle \beta {\left(w\right)}_t-d_{\epsilon }diva(x,Dw)+r_{\epsilon }g\left(x,\beta \left(w\right)\right)=f,}$ où $\mathbf{a}$ 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 ${\displaystyle \epsilon \to 0}$. Dans le cas où, le coefficient de réaction est très rapide, nous étudions le comportement asymptotique lorsque ${\displaystyle t\to \infty }$ de la solution du problème d’obstacle afin de caractériser la donnée initiale du problème limite.

In this paper, we study the competition between the diffusion and the reaction for the problem of type ${\displaystyle \beta {\left(w\right)}_t-d_{\epsilon }diva(x,Dw)+r_{\epsilon }g\left(x,\beta \left(w\right)\right)=f,}$ 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 ${\displaystyle \epsilon \to 0}$. In the case when, the reaction coefficient is very fast, we study the asymptotic behavior as ${\displaystyle t\to \infty }$ of the solution of the obstacle problem to characterize the initial data for the limit problem.

[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 | Zbl

[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 | Zbl

[3] Ph. Benilan, L. Boccardo, Th. Gallouet, R. Gariepy, M. Pierre and J .L. Vazquez, An $L^1$-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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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