Geodesic
On~the partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 29, Tome 249 (1997), pp. 20-39

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We consider quasilinear nondiagonal parabolic systems with quadratic growth on the gradient in a parabolic cylinder $Q$. Under Dirichlet and Neumann boundary conditions partial Hölder continuity up to the lateral surface of $Q$ of solutions $u\in W_2^{1,1} (Q)\cap L^\infty(Q)$ is proved. Hausdorff dimension of a singular set is estimated. In the proof we get of the maximum principle theorem for corresponding model linear problems. 
@article{ZNSL_1997_249_a1,
     author = {A. A. Arkhipova},
     title = {On~the partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {20--39},
     publisher = {mathdoc},
     volume = {249},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_249_a1/}
}
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A. A. Arkhipova. On~the partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 29, Tome 249 (1997), pp. 20-39. http://geodesic.mathdoc.fr/item/ZNSL_1997_249_a1/