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Exceptional sets for solutions to quasilinear parabolic equations in weighted Sobolev spaces. (Russian) Zbl 1015.35022

The article is devoted to studying the question of elimination of singularity for bounded solutions to a quasilinear parabolic equation of the form \[ u_t - \operatorname{div }A(x,t,u,\nabla_{x} u) + B(x,t,u,\nabla_{x} u) = 0 \] for \((x,t)\in Q_T = \Omega\times (0,T)\), where \(\Omega \subset \mathbb R^n\) is a bounded domain.
The main result of the article reads as follows: Let \(e\) be a compact subset of \(Q_T\) having zero \((p,\mu)\)-capacity. If \(u \in W_{p,\text{loc}}^{1,0}(Q_T \setminus e,\mu)\cap L_{\infty}(Q_T)\) satisfies the equation on the set \(Q_T \setminus e\), then there exists a unique continuation \(\tilde u \in W_{p,\text{loc}}^{1,0}(Q_T,\mu)\) of the function \(u\) such that \(\tilde u\) is a solution to the equation.

MSC:

35B60 Continuation and prolongation of solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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