Geodesic
Global existence for quasilinear diffusion equations in isotropic nondivergence form
Arendt, Wolfgang1  ; Chill, Ralph2 
1 Institut für Angewandte Analysis, Universität Ulm, D-89069 Ulm, Germany
2 Université Paul Verlaine - Metz, Laboratoire de Mathématiques et Applications de Metz et CNRS, UMR 7122, Bât. A, Ile du Saulcy, 57045 Metz Cedex 1, France
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 523-539

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We consider the quasilinear parabolic equation u t - β ( t , x , u , ∇ u ) Δ u = f ( t , x , u , ∇ u ) in a cylindrical domain, together with initial-boundary conditions, where the quasilinearity operates on the diffusion coefficient of the Laplacian. Under suitable conditions we prove global existence of a solution in the energy space. Our proof depends on maximal regularity of a nonautonomous linear parabolic equation which we use to provide us with compactness in order to apply Schaefer’s fixed point theorem.

Classification : 35K15, 35A05, 35K55

Arendt, Wolfgang 1 ; Chill, Ralph 2

1 Institut für Angewandte Analysis, Universität Ulm, D-89069 Ulm, Germany
2 Université Paul Verlaine - Metz, Laboratoire de Mathématiques et Applications de Metz et CNRS, UMR 7122, Bât. A, Ile du Saulcy, 57045 Metz Cedex 1, France 
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     title = {Global existence for quasilinear diffusion equations in isotropic nondivergence form},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {523--539},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {3},
     year = {2010},
     mrnumber = {2722654},
     zbl = {1223.35202},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_3_523_0/}
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Arendt, Wolfgang; Chill, Ralph. Global existence for quasilinear diffusion equations in isotropic nondivergence form. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 523-539. http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_3_523_0/