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Unbeschränkte Randwerte bei parabolischen Differentialgleichungen

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Abstract

In this paper we will prove the existence of classical solutions u for a quasilinear parabolic differential equation of type

$$(1)u_t = \sum\limits_{i = 1}^n {\tfrac{d}{{dx_i }}a_i (x,t,u,u_x )} + a(x,t,u,u_x )$$

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.

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Authors
  1. Gerhard Schleinkofer
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Institut für Angewandte Mathematik der Universität Saarstraße 21 65 Mainz

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Schleinkofer, G. Unbeschränkte Randwerte bei parabolischen Differentialgleichungen. Manuscripta Math 5, 373–384 (1971). https://doi.org/10.1007/BF01367771

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  • DOI: https://doi.org/10.1007/BF01367771

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