Abstract
In this paper we will prove the existence of classical solutions u for a quasilinear parabolic differential equation of type
in a cylindrical domain G, whereby unbounded boundary values g may be given on the parabolic boundary Rp of G. In general, u is unbounded and ux∉L2(G). Under certain conditions (see Satz 3) we nevertheless get a reasonable boundary-behavior of u, that is:\(u(Q) \to g(\bar Q)\) when g is continuous in\(\bar Q \in {\text{R(}}Q \to \bar Q{\text{)}}\), and u(Γ)→g in the L2-sense for Γ→Rp where Γ means a parallel surface to Rp.
Similar content being viewed by others
Literatur
Aronson, D. G., Serrin, J.: Local behavior of solutions of quasilinear parabolic equations. Arch. Rat. Mech. Anal. 25, 2, 81–122 (1967).
Browder, F. E.: Existence theorems for nonlinear partial differential equations. Proc. Symp. Pure Math. AMS, vol. 16, 1–60 (1970).
Cimmino, G.: Nuovo tipo di condizione al contorno e nuovo metodo di trattazione per il problema generalizzato di Dirichlet. Rend. Circ. Mat. Palermo 61, 177–220 (1937).
Ladyženskaja, O. A., Ural'tseva, N. N., Solonnikov, V. A.: Linear and quasilinear equations of parabolic type. AMS, Trans. Math. Monogr. 23 (1968).
Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires; Dunod, Ganthier-Villars, Paris (1969).
Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Masson et Cie,, Editeurs, Paris; Academia, Editeurs, Prague (1967).
Oleinik, O. A., Krushkov, S. N.: Quasilinear second order parabolic equations with many independent variables. Uspehi Mat. Nauk 16 (1961); Russian Math. Surveys 16, 105–146 (1961).
Trudinger, N. S.: The first initial boundary value problem for quasilinear parabolic equations, AMS Notices 14, 241 (1967).
—: Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math. 21, 205–226 (1968).
Vito, L. De: Sulle funzioni ad integrale di Dirichlet finito. Ann. Sc. Norm. Sup. Pisa 12 (1958), 55–127.
Walter, W.: Differential and integral inequalities. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55. Berlin-Heidelberg-New York: Springer 1970.
Additional information
Institut für Angewandte Mathematik der Universität Saarstraße 21 65 Mainz
Rights and permissions
About this article
Cite this article
Schleinkofer, G. Unbeschränkte Randwerte bei parabolischen Differentialgleichungen. Manuscripta Math 5, 373–384 (1971). https://doi.org/10.1007/BF01367771
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01367771