Classical Solution To The First Initial-Boundary Value Problem Of Parabolic Hessian Equation

In this paper, we study classical solution to the first initial-boundaryvalue problem of parabolic Hessian equation:where is a bounded, uniformly (k-l)-convex domain in Rn with smooth boundary . denote the eigenvalues of the Hessian matrix of second derivatives , while Sk isthe elementary symmetric function, for Problem (1) was treated in [5] under different conditions, for the case, which discussed the existence and uniqueness of classical solutions.Later, [3] discussed the existence and uniqueness of classical solutions for problem (1) when f satisfies the structure conditions in [l],and their results included those of [5}.f of (2) dosen't satisfy the structure condition (7) in [1](see [2]),so the function f discussed in [3] precludes the case discussed in this paper.This paper uses compare principle to show that there exists at most one of classical solution for (1), while the existance of solution is obtained through continuous method.To get the required a priori estimates except the double normal derivatives,we adopt the method in [3], and the double normal derivatives on dQ are achieved by barrier constructions and applying skill of [2].