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Zeitschrift für Analysis und ihre Anwendungen

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Volume 3, Issue 5, 1984, pp. 413–423
DOI: 10.4171/ZAA/118

Published online: 1984-10-31

Ein Randwertproblem für eine nichtlineare Gleichung gemischtenTyps im $\mathbb R^3$

Andreas Müller-Rettkowski[1]

(1) Karlsruhe Institute of Technology (KIT), Germany

A boundary value problem for the equation $Tu = Lu — u |u|^p = f(x, u), p > 0$, is studied in a simply connected bounded domain $G$ of $\mathbb R^3$. Here $L$ denotes a linear second order differential operator which is elliptic, parabolic or hyperbolic if $x_3 > 0, x_3 = 0$ or $x_3 < 0$, respectively. The boundary of 0 is formed by a non-characteristic and by two characteristic surfaces. The boundary value problem to be solved is to find a solution of the equation in 0 which assumes zero data on the non-characteristic and on one of the characteristic boundary surfaces. It is proved that this problem has a generalized solution belonging to $L^{p+2}$ and to a Sobolew space with weight. Using apriori estimates the solubility of a sequence of approximate problems is shown whose solutions turn out to converge towards a solution of the boundary value problem in question.

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Müller-Rettkowski Andreas: Ein Randwertproblem für eine nichtlineare Gleichung gemischtenTyps im $\mathbb R^3$. Z. Anal. Anwend. 3 (1984), 413-423. doi: 10.4171/ZAA/118