[Diffusions non linéaires et constantes optimales dans des inégalités de type Sobolev : comportement asymptotique d'équations faisant intervenir le p-Laplacien]
Nous étudions le comportement asymptotique des solutions positives ou nulles de : ut=Δpum à l'aide d'une estimation d'entropie qui repose sur l'utilisation d'une sous-famille des inégalités de Gagliardo–Nirenberg – ou, dans le cas limite m=(p−1)−1, d'une inégalité de Sobolev logarithmique dans W1,p – pour laquelle on connait des fonctions optimales.
We study the asymptotic behaviour of nonnegative solutions to: ut=Δpum using an entropy estimate based on a sub-family of the Gagliardo–Nirenberg inequalities – or, in the limit case m=(p−1)−1, on a logarithmic Sobolev inequality in W1,p – for which optimal functions are known.
Accepté le :
Publié le :
Manuel Del Pino 1 ; Jean Dolbeault 2
@article{CRMATH_2002__334_5_365_0,
author = {Manuel Del Pino and Jean Dolbeault},
title = {Nonlinear diffusions and optimal constants in {Sobolev} type inequalities: asymptotic behaviour of equations involving the $ \mathbf{p}${-Laplacian}},
journal = {Comptes Rendus. Math\'ematique},
pages = {365--370},
publisher = {Elsevier},
volume = {334},
number = {5},
year = {2002},
doi = {10.1016/S1631-073X(02)02225-2},
language = {en},
}
TY - JOUR
AU - Manuel Del Pino
AU - Jean Dolbeault
TI - Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the $ \mathbf{p}$-Laplacian
JO - Comptes Rendus. Mathématique
PY - 2002
SP - 365
EP - 370
VL - 334
IS - 5
PB - Elsevier
DO - 10.1016/S1631-073X(02)02225-2
LA - en
ID - CRMATH_2002__334_5_365_0
ER -
%0 Journal Article
%A Manuel Del Pino
%A Jean Dolbeault
%T Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the $ \mathbf{p}$-Laplacian
%J Comptes Rendus. Mathématique
%D 2002
%P 365-370
%V 334
%N 5
%I Elsevier
%R 10.1016/S1631-073X(02)02225-2
%G en
%F CRMATH_2002__334_5_365_0
Manuel Del Pino; Jean Dolbeault. Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the $ \mathbf{p}$-Laplacian. Comptes Rendus. Mathématique, Volume 334 (2002) no. 5, pp. 365-370. doi : 10.1016/S1631-073X(02)02225-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02225-2/
[1] On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations, Comm. Partial Differential Equations, Volume 26 (2001) no. 1–2, pp. 43-100
[2] Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Asymptotic Methods in Kinetic Theory, Volume 119 (1999), pp. 1-91 (Preprint TMR)
[3] Asymptotic L1-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., Volume 49 (2000), pp. 113-141
[4] Del Pino M., Dolbeault J., Generalized Sobolev inequalities and asymptotic behaviour in fast diffusion and porous media problems, Preprint Ceremade no. 9905, 1999, pp. 1–45
[5] Del Pino M., Dolbeault J., Best constants for Gagliardo–Nirenberg inequalities and application to nonlinear diffusions, Preprint Ceremade no. 0119, 2001, pp. 1–25, J. Math. Pures Appl. (to appear)
[6] Del Pino M., Dolbeault J., General logarithmic and Gagliardo–Nirenberg inequalities with best constants, Preprint Ceremade no. 0120, 2001, pp. 1–12. Preprint CMM-B-01/06-38, 2001, pp. 1–11
[7] Del Pino M., Dolbeault J., Asymptotic behaviour of nonlinear diffusions (in preparation)
[8] Degenerate Parabolic Equations, Springer-Verlag, New York, 1993
[9] The asymptotic behaviour of gas in a n-dimensional porous medium, Trans. Amer. Math. Soc., Volume 262 (1980) no. 2, pp. 551-563
[10] Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, Volume 4 (1988) no. 2, pp. 339-354
[11] The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, Volume 26 (2001) no. 1–2, pp. 101-174
[12] Uniqueness for ground states of quasilinear elliptic equations, Indiana Univ. Math. J., Volume 49 (2000) no. 3, pp. 897-923
[13] Sur l'inégalité logarithmique de Sobolev, C. R. Acad. Sci. Paris, Série I, Volume 324 (1997), pp. 689-694
Cité par Sources :
Commentaires - Politique