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Zeitschrift für Analysis und ihre Anwendungen


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Volume 15, Issue 1, 1996, pp. 159–180
DOI: 10.4171/ZAA/693

Published online: 1996-03-31

A Necessary Condition to Regularity of a Boundary Point for a Degenerate Quasilinear Parabolic Equation

Salvatore Leonardi[1] and I.I. Skrypnik[2]

(1) Università degli Studi di Catania, Italy
(2) Academy of Sciences of Ukraine, Donetsk, Ukraine

We shall study the behaviour of solutions of the equation $$v(x) \frac{\partial u}{\partial t} – \sum^n_{i=l} \frac{\partial}{\partial x_i} a_i (x, t, u \frac{\partial u}{\partial x}) = a_0 (x, t, u \frac{\partial u}{\partial x}) \\ ((x,t) \in Q_T = \Omega \times (0, T))$$ at a point $(x_0,t_0) \in S_T = \partial \Omega \times (0, T)$. Indeed we establish a necessary condition to the regularity of a boundary point of the cylindrical domain $Q_T$ extending the analogous result from paper [13] to the degenerate case. The degeneration is given by weights (depending on the space variable) from a suitable Muchenhoupt class. It is important to note that the coefficients of the equation depend on time too.

Keywords: Degenerate nonlinear parabolic equations, regularity at boundary points

Leonardi Salvatore, Skrypnik I.I.: A Necessary Condition to Regularity of a Boundary Point for a Degenerate Quasilinear Parabolic Equation. Z. Anal. Anwend. 15 (1996), 159-180. doi: 10.4171/ZAA/693