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Unilateral problems with convex constraints for parabolic quasilinear operators. (English. Russian original) Zbl 0677.35060
Sib. Math. J. 29, No. 5, 846-857 (1988); translation from Sib. Mat. Zh. 29, No. 5(171), 198-211 (1988).
A priori estimates of solutions are obtained for a quasilinear parabolic problem with unilateral constraints imposed on the solution’s gradient. A set that defines these constraints may be unbounded and (in case of an operator with a linear principal part) non-strictly convex. The derived estimates provide the solvability of the problem studied in $$L_{\infty}(0,T;W^ 2_{p,loc}(\Omega)),$$ $$\Omega \subset R^ n$$, $$0<t<T$$. Also are established conditions under which the solution belongs to $$L_{\infty}(0,T;W^ 2_ p(\Omega))\cap C^{0,1}({\bar \Omega}),$$ $$1<p<\infty$$ or $$L_{\infty}(0,T;W^ 2_{\infty}(\Omega))$$.
Reviewer: I.Zino
MSC:
 35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35K20 Initial-boundary value problems for second-order parabolic equations
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References:
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