Global existence for quasilinear diffusion equations in isotropic nondivergence form

Arendt, Wolfgang; Chill, Ralph

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 $u_t-\beta (t,x,u,\nabla u)\Delta u=f(t,x,u,\nabla u)$ 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

[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. | MR | EuDML | 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. | MR | EuDML

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