Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the 𝐩-Laplacian
[Diffusions non linéaires et constantes optimales dans des inégalités de type Sobolev : comportement asymptotique d'équations faisant intervenir le p-Laplacien]
Del Pino, Manuel1 ; Dolbeault, Jean2
1 DIM & CMM (UMR CNRS 2071), FCFM, Univ. de Chile, Casilla 170 Correo 3, Santiago, Chile
2 Ceremade (UMR CNRS 7534), Univ. Paris IX-Dauphine, Pl. de Lattre de Tassigny, 75775 Paris cedex 16, France
Comptes Rendus. Mathématique, Tome 334 (2002) no. 5, pp. 365-370.

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.

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Accepté le :
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DOI : 10.1016/S1631-073X(02)02225-2
Del Pino, Manuel 1 ; Dolbeault, Jean 2

1 DIM & CMM (UMR CNRS 2071), FCFM, Univ. de Chile, Casilla 170 Correo 3, Santiago, Chile
2 Ceremade (UMR CNRS 7534), Univ. Paris IX-Dauphine, Pl. de Lattre de Tassigny, 75775 Paris cedex 16, France 
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     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},
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Del Pino, Manuel; Dolbeault, Jean. Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the $ \mathbf{p}$-Laplacian. Comptes Rendus. Mathématique, Tome 334 (2002) no. 5, pp. 365-370. doi : 10.1016/S1631-073X(02)02225-2. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02225-2/

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