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One-sided problems for parabolic quasilinear operators. (English. Russian original) Zbl 0658.35052
Sov. Math., Dokl. 34, 327-330 (1987); translation from Dokl. Akad. Nauk SSSR 290, 549-552 (1986).
Consider a so-called unilateral problem for a parabolic quasilinear operator L, which is defined as $Lu=-v_ t+a_{ij}(x,t,v_ x)v_{x_ ix_ j}+a(x,t,v,v_ x).$ To find a function $$u\in \cap_{1<p<\infty}L_{\infty}(0,T;W^ 2_{p,loc}(\Omega))\cap C^{0,1}(\bar Q)$$ satisfying the conditions
(1) Lu$$\geq 0$$ almost everywhere in Q;
(2) $$F_ m(x,t,u,u_ x)\leq 0$$ in Q for $$m=1,...,N$$ if at the point (x,t)$$\in Q$$ the inequalities
(3) $$F_ m(x,t,u,u_ x)<0$$ for all $$m=1,...,N$$ hold, then in a neighborhood of this point $$Lu=0$$ almost everywhere; and
(4) $$u=0$$ on $$\Gamma =(\partial \Omega \times [0,T])\cup (\Omega \times \{0\}).$$
Many authors studied this problem [cf. Lawrence C. Evans, Commun. Partial Differ. Equations 4, 555-572 (1979; Zbl 0448.35036)]. Here the solvability and limiting smoothness of a solution of problem (1)-(4) are established. A class of convex constraints is here considered.
Reviewer: J.H.Tian

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### Keywords:
unilateral; quasilinear; solvability; smoothness; convex constraints