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

We consider the general degenerate parabolic equation :

u t - Δ b ( u ) + d i v F ˜ ( u ) = f in Q = ] 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.

Nous considérons l’équation parabolique dégénérée général :

u t - Δ b ( u ) + d i v F ˜ ( u ) = f dans Q = ] 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.

DOI: 10.5802/afst.1194
Maliki, Mohamed 1; Ouedraogo, Adama 2

1 Department of Mathematics BP 146, Hassan II University Mohammedia (Morocco)
2 Department of Mathematics 03 BP 7021 University of Ouagadougou 03 (Burkina Faso)
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Maliki, Mohamed; Ouedraogo, Adama. Renormalized solution for nonlinear degenerate problems in the whole space. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 3, pp. 597-611. doi : 10.5802/afst.1194. http://archive.numdam.org/articles/10.5802/afst.1194/

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