Geodesic
Existence of solutions of parabolic variational inequalities with one-sided restrictions
Matematičeskie zametki, Tome 77 (2005) no. 3, pp. 460-476.

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We prove sufficient conditions for the existence of a solution of a “strong” nonlinear variational inequality of parabolic type. The theory can be used for solving parabolic equations with one-sided boundary conditions. As an example, we prove the existence of a solution of a strong parabolic variational inequality with $p$-Laplacian in the Sobolev space $L_p(0,T;W_p^1(\Omega))$, $p\in[2,\infty)$. 
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O. V. Solonukha. Existence of solutions of parabolic variational inequalities with one-sided restrictions. Matematičeskie zametki, Tome 77 (2005) no. 3, pp. 460-476. http://geodesic.mathdoc.fr/item/MZM_2005_77_3_a11/

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