Rozhkovskaya, T. N. 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 Keywords:existence; A priori estimates; quasilinear; unilateral constraints; linear principal part; non-strictly convex; solvability PDF BibTeX XML Cite \textit{T. N. Rozhkovskaya}, Sib. Math. J. 29, No. 5, 846--857 (1988; Zbl 0677.35060); translation from Sib. Mat. Zh. 29, No. 5(171), 198--211 (1988) Full Text: DOI References: [1] L. C. Evans, ?A second order elliptic equation with gradient constraint,? Commun. Part. Different. Equat.,4, No. 5, 361-371 (1981). [2] T. N. Rozhkovskaya, ?Unilateral problems for elliptic operators with convex constraints on the gradient of the solution,? Sib. Mat. Zh.,26, No. 3, 134-147;26, No. 5, 150-158 (1986). [3] T. N. Rozhkovskaya, ?Unilateral problems for parabolic quasilinear operators,? Dokl. Akad. Nauk SSSR,290, No. 3, 549-552 (1986). [4] H. Ishii and S. Koike, ?Boundary regularity and uniqueness for an elliptic equation with gradient constraint,? Commun. Part. Different. Equat.,8, No. 4, 317-346 (1983). · Zbl 0538.35012 · doi:10.1080/03605308308820271 [5] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967). · Zbl 0164.12302 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.