Équations aux dérivées partielles
Une estimation de type Aronson–Bénilan
[An Aronson–Bénilan estimate]
Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 991-996.

We consider the equation utϕ(u), where ϕ∈C3(0,∞) is increasing. Under the condition ν·″(s)/ϕ′(s)⩾γ for some γ>0 and ν∈{−1;1}, we prove the estimate ν·du/dt⩾−u/γt. This result improves the estimates given by M.G. Crandall and M. Pierre (in J. Funct. Anal. 45 (1982) 194–212) for this equation.

Nous considérons l'équation utϕ(u), où ϕ∈C3(0,∞) est croissante. Sous l'hypothèse ν·″(s)/ϕ′(s)⩾γ pour un γ>0 et ν∈{−1;1}, nous montrons l'estimation ν·du/dt⩾−u/γt. Ce résultat améliore les estimations donnée par M.G. Crandall et M. Pierre (dans J. Funct. Anal. 45 (1982) 194–212) pour cette équation.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00255-3
Chasseigne, Emmanuel 1

1 Université de Tours, parc de Grandmont, 37200 Tours, France
@article{CRMATH_2003__336_12_991_0,
     author = {Chasseigne, Emmanuel},
     title = {Une estimation de type {Aronson{\textendash}B\'enilan}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {991--996},
     publisher = {Elsevier},
     volume = {336},
     number = {12},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00255-3},
     language = {fr},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00255-3/}
}
TY  - JOUR
AU  - Chasseigne, Emmanuel
TI  - Une estimation de type Aronson–Bénilan
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 991
EP  - 996
VL  - 336
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00255-3/
DO  - 10.1016/S1631-073X(03)00255-3
LA  - fr
ID  - CRMATH_2003__336_12_991_0
ER  - 
%0 Journal Article
%A Chasseigne, Emmanuel
%T Une estimation de type Aronson–Bénilan
%J Comptes Rendus. Mathématique
%D 2003
%P 991-996
%V 336
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00255-3/
%R 10.1016/S1631-073X(03)00255-3
%G fr
%F CRMATH_2003__336_12_991_0
Chasseigne, Emmanuel. Une estimation de type Aronson–Bénilan. Comptes Rendus. Mathématique, Volume 336 (2003) no. 12, pp. 991-996. doi : 10.1016/S1631-073X(03)00255-3. http://www.numdam.org/articles/10.1016/S1631-073X(03)00255-3/

[1] Aronson, D.G.; Bénilan, P. Régularité des solutions de l'équation des milieux poreux dans N , C. R. Acad. Sci. Paris, Volume 288 (1979), pp. 103-105

[2] Crandall, M.G.; Pierre, M. Regularizing effects for utϕ(u), Trans. Amer. Math. Soc., Volume 274 (1982), pp. 159-168

[3] Crandall, M.G.; Pierre, M. Regularizing effects for ut=(u) in L1, J. Funct. Anal., Volume 45 (1982), pp. 194-212

[4] Ladyzhenskaja, O.A.; Solonnikov, V.A.; Uralceva, N.N. Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Transl., 23, 1968

[5] L.A. Peletier, The porous media equation, Applications of nonlinear analysis in the physical sciences, Pap. Workshop, Bielefeld 1979, 1981, pp. 229–241

[6] Vazquez, J.L. Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures Appl., Volume 71 (1992), pp. 503-526

[7] Vazquez, J.L. An introduction to the mathematical theory of the porous medium equation, Shape Optimization and free Boundaries, Proc. NATO ASI, Sémin. Math. Supér., Montréal/Canada 1990, NATO Adv. Sci. Inst. Ser. C, 380, 1992, pp. 347-389

Cited by Sources: