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The Petrovskiĭ criterion and barriers for degenerate and singular \(p\)-parabolic equations. (English) Zbl 1387.35376

The counterpart of Petrovski’s criterion for the regularity of the latest boundary point is generalized from the heat equation to the evolutionary \(p\)-Laplace equation \[ {\partial u\over\partial t}=\nabla\cdot(|\nabla u|^{p-2}\nabla u),\qquad u= u(x_1,\dots, x_n,t),\tag{1} \] in the domain \[ \Omega= \{(x,t)\mid|x|<\zeta(t),\;T_0<t<0\}. \] Here \(\zeta(t)\) is a positive continuous function.
In the case \(p>2\) the following result is obtained:
\(\bullet\)
\(\lim_{t\to 0-} (-t)^{-1/p}\zeta(t)= 0\Rightarrow (0,0)\) is regular,
\(\bullet\)
\(\liminf_{t\to 0-} (-t)^{-1/p}\zeta(t)> 0\Rightarrow (0,0)\) is irregular.
In the case \(1<p<2\) the authors construct a domain for which the origin is an irregular boundary point. They exhibit one classical barrier function at the origin. This indicates that a whole family of barriers is needed to guarantee regularity.
The proofs are based on elaborate calculations. The trivial observation that the constant \(k\) can be scaled away from the equation \[ {\partial u\over\partial t}=k\nabla\cdot(|\nabla u|^{p-2}\nabla u)\tag{2} \] when \(p\neq 2\) is used. (Thus equations (1) and (2) are similar if \(p\neq 2\).)

MSC:

35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
35K67 Singular parabolic equations
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
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References:

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