## 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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