Advances in Differential Equations

On the Wiener test for degenerate parabolic equations with non-standard growth condition

Igor I. Skrypnik

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We investigate the continuity of solutions for general nonlinear parabolic equations of the form \begin{equation*} u_t-\sum_{i=1}^n\frac{\partial}{\partial x_i}\Bigl(\Bigl|\frac{\partial u}{\partial x_i}\Bigr|^{p(x)-2}\frac{\partial u}{\partial x_i}\Bigr)=0,\qquad 2 < p_1\leq p(x)\leq p_2 \end{equation*} near a nonsmooth boundary of a cylindrical domain. We prove the sufficient and necessary condition for regularity of a boundary point in terms of the $p(x)$-capacity.

Article information

Adv. Differential Equations, Volume 13, Number 3-4 (2008), 229-272.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K65: Degenerate parabolic equations
Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations


Skrypnik, Igor I. On the Wiener test for degenerate parabolic equations with non-standard growth condition. Adv. Differential Equations 13 (2008), no. 3-4, 229--272.

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