JournalszaaVol. 15, No. 1pp. 159–180

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

  • Salvatore Leonardi

    Università degli Studi di Catania, Italy
  • I.I. Skrypnik

    Academy of Sciences of Ukraine, Donetsk, Ukraine
A Necessary Condition to Regularity of a Boundary Point for a Degenerate Quasilinear Parabolic Equation cover
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Abstract

We shall study the behaviour of solutions of the equation

v(x)uti=lnxiai(x,t,uux)=a0(x,t,uux)((x,t)QT=Ω×(0,T))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 (x0,t0)ST=Ω×(0,T)(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 QTQ_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

Salvatore 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