Geodesic
On the existence of smooth solutions for parabolic sistems with convex constraints on the boundary
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 20, Tome 171 (1989), pp. 5-11

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The existence of the smooth solution for the convex boundary — constrained problem is prooved. Parabolic operator has diagonal form and quadratic growth for gradient. This solution has maximal possible regularity for the boundary obstacle problems. 
@article{ZNSL_1989_171_a0,
     author = {A. A. Arkhipova and N. N. Uraltseva},
     title = {On the existence of smooth solutions for parabolic sistems with convex constraints on the boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--11},
     publisher = {mathdoc},
     volume = {171},
     year = {1989},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1989_171_a0/}
}
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A. A. Arkhipova; N. N. Uraltseva. On the existence of smooth solutions for parabolic sistems with convex constraints on the boundary. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 20, Tome 171 (1989), pp. 5-11. http://geodesic.mathdoc.fr/item/ZNSL_1989_171_a0/