Partial Differential Equations
On nonlinear diffusion problems with strong degeneracy
Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 569-572.

In this Note, we study the ‘triply’ degenerate problem: b(v)tΔg(v)+divΦ(v)=f on Q:=(0,T)×Ω, b(v(0,))=b(v0) on Ω and g(v)=g(a) ‘on some part of the boundary’ (0,T)×Ω, in the case of continuous nonhomogenous and nonstationary boundary data a. The functions b,g are assumed to be continuous nondecreasing and to verify the normalisation condition b(0)=g(0)=0 and the range condition R(b+g)=R. Using monotonicity and penalization methods, we prove existence of a weak entropy solution in the spirit of F. Otto (1996).

Dans cette Note, on étudie le problème triplement dégénéré : b(v)tΔg(v)+divΦ(v)=f sur Q:=(0,T)×Ω, b(v(0,))=b(v0) dans Ω et g(v)=g(a) « sur une partie de la frontière » (0,T)×Ω, dans le cas d'une donnée a continue non homogène et non stationnaire sur le bord. Les fonctions b,g sont supposées être continues croissantes, vérifiant la condition de normalisation : b(0)=g(0)=0 et de surjectivité R(b+g)=R. En utilisant des méthodes de monotonie et de pénalisation, on prouve l'existence d'une solution entropique au sens de F. Otto (1996).

Published online:
DOI: 10.1016/j.crma.2006.09.030
Ammar, Kaouther 1

1 Institut für Mathematik, TU Berlin, Strasse des 17 juni 135, 10625 Berlin, Germany
     author = {Ammar, Kaouther},
     title = {On nonlinear diffusion problems with strong degeneracy},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {569--572},
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     year = {2006},
     doi = {10.1016/j.crma.2006.09.030},
     language = {en},
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Ammar, Kaouther. On nonlinear diffusion problems with strong degeneracy. Comptes Rendus. Mathématique, Volume 343 (2006) no. 9, pp. 569-572. doi : 10.1016/j.crma.2006.09.030.

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