One-sided problems for parabolic quasilinear operators. (English. Russian original) Zbl 0658.35052

Rozhkovskaya, T. N.

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