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Parabolic type Monge-Ampère equation with zero initial boundary value. (English) Zbl 1041.35025
The authors prove the existence and uniqueness of generalized convex-monotone solutions, as formulated by R. Wang and G. Wang [J. Partial Differ. Equations 6, No. 3, 237–254 (1993; Zbl 0811.35053)], of the problem \[ -u_t\det D^2_xu = f(x,t) \quad\text{in}\quad Q=\Omega\times(0,T], \qquad u=0 \quad\text{on}\quad \partial_pQ, (*) \] where \(\Omega\) is a bounded convex domain in \({\mathbb R}^n\), \(\partial_pQ\) denotes the parabolic boundary of \(Q\), and \(f\) is a nonnegative bounded measurable function. Closely related results are contained in the papers of L. Chen, G. Wang and S. Lian [J. Partial Differ. Equations 14, No. 2, 149–162 (2001; Zbl 0990.35034); J. Differ. Equations 186, No. 2, 558–571 (2002; Zbl 1014.35042)], and of J. L. Spiliotis [J. Math. Anal. Appl. 163, No. 2, 484–511(1992; Zbl 0753.35044); Nonlinear Stud. 4, No. 2, 233–255 (1997; Zbl 0883.35058)].

35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)