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Renormalized solution for nonlinear degenerate problems in the whole space
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 3, pp. 597-611.

Nous considérons l’équation parabolique dégénérée général :

ut-Δb(u)+divF˜(u)=fdansQ=]0,T[×N,T>0.

Nous supposons que le flux F ˜ est continu, b est continue et croissante au sens large et les deux fonctions ne sont pas nécessairement lipschitziennes. Nous prouvons l’existence de solution renormalisée du problème de Cauchy associé à cette équation avec des données (terme source et condition initiale) dans L 1 . Nous établissons l’unicité de cette solution sous une condition dite de structure du type F ˜(r)=F(b(r)) et sous une hypothèse sur le module de continuité de b. La nouveauté dans le travail vient du fait que Ω= N , u 0 , fL 1 , b, F ˜ ne sont pas des fonctions nécessairement lipschitziennes et les techniques sont différentes de celles développées dans les travaux antérieurs.

We consider the general degenerate parabolic equation :

ut-Δb(u)+divF˜(u)=finQ=]0,T[×N,T>0.

We suppose that the flux F ˜ is continuous, b is nondecreasing continuous and both functions are not necessarily Lipschitz. We prove the existence of the renormalized solution of the associated Cauchy problem for L 1 initial data and source term. We establish the uniqueness of this type of solution under a structure condition F ˜(r)=F(b(r)) and an assumption on the modulus of continuity of b. The novelty of this work is that Ω= N , u 0 , fL 1 , b, F ˜ are not Lipschitz functions and the techniques are different from those developed in the previous works.

Reçu le : 2006-07-24
Accepté le : 2007-11-29
Publié le : 2010-12-06
DOI : https://doi.org/10.5802/afst.1194
@article{AFST_2008_6_17_3_597_0,
     author = {Mohamed Maliki and Adama Ouedraogo},
     title = {Renormalized solution for nonlinear degenerate problems in the whole space},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 17},
     number = {3},
     year = {2008},
     pages = {597-611},
     doi = {10.5802/afst.1194},
     zbl = {1173.35565},
     mrnumber = {2488233},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2008_6_17_3_597_0/}
}
Mohamed Maliki; Adama Ouedraogo. Renormalized solution for nonlinear degenerate problems in the whole space. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 3, pp. 597-611. doi : 10.5802/afst.1194. https://afst.centre-mersenne.org/item/AFST_2008_6_17_3_597_0/

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