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.

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
@article{ASNSP_2010_5_9_3_523_0,
     author = {Arendt, Wolfgang and Chill, Ralph},
     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://archive.numdam.org/item/ASNSP_2010_5_9_3_523_0/}
}
TY  - JOUR
AU  - Arendt, Wolfgang
AU  - Chill, Ralph
TI  - Global existence for quasilinear diffusion equations in isotropic nondivergence form
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2010
SP  - 523
EP  - 539
VL  - 9
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2010_5_9_3_523_0/
LA  - en
ID  - ASNSP_2010_5_9_3_523_0
ER  - 
%0 Journal Article
%A Arendt, Wolfgang
%A Chill, Ralph
%T Global existence for quasilinear diffusion equations in isotropic nondivergence form
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2010
%P 523-539
%V 9
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2010_5_9_3_523_0/
%G en
%F ASNSP_2010_5_9_3_523_0
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://archive.numdam.org/item/ASNSP_2010_5_9_3_523_0/

[1] H. Amann, Maximal regularity and quasilinear parabolic boundary value problems, In: “Recent Advances in Elliptic and Parabolic Problems”, World Sci. Publ., Hackensack, NJ, 2005, 1–17. | MR | Zbl

[2] H. Amann, Quasilinear parabolic problems via maximal regularity, Adv. Differential Equations 10 (2005), 1081–1110. | MR | Zbl

[3] H. Berestycki, L. Nirenberg, and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994), 47–92. | Zbl

[4] Th. Cazenave and A. Haraux, “An Introduction to Semilinear Evolution Equations”, Oxford Lecture Series in Mathematics and its Applications, Vol. 13, Oxford University Press, New York, 1998. | MR | Zbl

[5] Ph. Clément and S. Li, Abstract parabolic quasilinear problems and application to a groundwater flow problem, Adv. Math. Sci. Appl. 3 (1994), 17–32. | MR | Zbl

[6] R. Dautray and J.-L. Lions, “Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques”, Vol. II, INSTN: Collection Enseignement, Masson, Paris, 1985. | MR | Zbl

[7] R. Dautray and J.-L. Lions, “Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques”, Vol. VIII, INSTN: Collection Enseignement, Masson, Paris, 1987. | MR | Zbl

[8] L. C. Evans, “Partial Differential Equations”, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. | MR

[9] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer Verlag, Berlin, Heidelberg, New York, 2001. | MR | Zbl

[10] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural^{\prime }ceva, “Linear and Quasilinear Equations of Parabolic Type”, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. | MR

[11] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. | MR | Zbl

[12] A. Lunardi, “Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Progress in Nonlinear Differential Equations and Their Applications, Vol. 16, Birkhäuser, Basel, 1995. | MR | Zbl

[13] J. Prüss, Maximal regularity for evolution equations in L p -spaces, Conf. Semin. Mat. Univ. Bari 285 (2002), 1–39. | MR

[14] H. H. Schaefer, Über die Methode der a priori-Schranken, Math. Ann. 129 (1955), 415–416. | EuDML | MR | Zbl

[15] R. E. Showalter, “Monotone Operators in Banach Space and Nonlinear Partial Differential Equations”, Mathematical Surveys and Monographs, vol. 49, American Mathematical Society, Providence, RI, 1997. | MR | Zbl

[16] A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 111 (1935), 767–776. | EuDML | MR

[17] A. Vitolo, Maximum principles for second-order parabolic equations, J. Partial Differential Equations, (4) 17 (2004), 289–302. | MR | Zbl