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

S. Leonardi; I.I. Skrypnik

doi:10.4171/zaa/693

JournalszaaVol. 15, No. 1pp. 159–180

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

S. Leonardi
Università degli Studi di Catania, Italy
I.I. Skrypnik
Academy of Sciences of Ukraine, Donetsk, Ukraine

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Abstract

We shall study the behaviour of solutions of the equation

v (x) \frac{\partial u}{\partial t} - i = l \sum n \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} = Ω \times (0, T))

at a point $(x_{0}, t_{0}) \in S_{T} = \partial Ω \times (0, T)$ . 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.

Cite this article

S. Leonardi, I.I. Skrypnik, A Necessary Condition to Regularity of a Boundary Point for a Degenerate Quasilinear Parabolic Equation. Z. Anal. Anwend. 15 (1996), no. 1, pp. 159–180

DOI 10.4171/ZAA/693