| In this paper we are concerned with the existence and uniqueness of the classical solutions to the initial and third boundary value problem for the more general parabolic Monge-Ampere equations as followswhereΩis a bounded uniformly convex domain in Rn,QT =Ω×(0,T];(?)pQT = (?)Ω×(0,T]∪Ω×{t = 0}, D2u = (Diju)is the Hessian of u, Dij = DiDj,Di = (?)/((?)x)i, i, j = 1, ...,n, v is the unit exterior normal at (x,t)∈(?)Ω×[0,T] to (?)Ω, which has been extended on QT to become a properly smooth vector field independent of t.α(x) > 0, (?)x∈Ω, and is properly smooth,σ=σ(x) = (σij(x))is an n×n matrix with smooth components. f(x, t),φ(x, t),ψ(x, t) are given properly smooth functions and satisfy some necessary compatibility conditions. u = u(x, t) is the unknown function.The study of this kind of problem can be traced back to the work studied by Ladyzhenkaya etc, which extends the work studied by Nirenberg etc on the problem of existence and uniqueness of solutions to the first boundary value problem for the elliptic Monge-Ampere equationto the problem on the existence and uniqueness of solutions to the initial and first boundary value problem for the parabolic Monge-Ampere equation Since Nirenberg etc also discussed the first boundary value problem for a more general elliptic Monge-Ampere equation of the formand P.L.Lions etc even studied the nonlinear Neumann problem for the above equation, which include the following the third boundary value condition for this equationTherefore it is natural for us to consider the problem (*) as an extension of the result of Ladyzhenkaya and Ivochkina.u = u(x,t) is called an admissible function of (*), ifIf an admissible function u = u(x, t) makes (*) become an identity, then this function u = u(x, t) is called the an admissible solution of (*).Obviously the equation (*) is of parabolic type for any admissible function u = u(x, t).For any admissible solutions, the following condition is necessary:We derive two structure conditions: The main result in the paper isTHEOREM:Assume thatΩis a bounded uniformly convex domain, and for someβ∈(0,1), (?)Ω∈C4+β,f(x,t)∈C2+β,1+β/2(QT),φ(x,t)∈C4+β,2+β/2(QT),φ(x,t)∈C4+β,2+β/2(qT) with compatibility conditions up to the second order satisfied. If besides condition (C1), one of the two conditions (C2) or (C2') holds, then the problem (*) has a unique admissible solution. u = u(x,t)∈C(4+β,2+β/2)(QT).To simplify the formulation, it is assumed above that:φ(x, t),φ(x, t) have been smoothly extended to all over QT withThe uniqueness part of the theorem is proved by a comparison theorem.The existence part of it is established by the method of continuity.It is well known that in order to use the method of continuity to establishing the existence of solutions, first of all one need to find such a family of problems with one parameter (?)∈[0,1], that the problem corresponding to (?) = 1 is just the problem in study, and that one can construct the solution to the problem corresponding to (?)= 0; after that one needs to show: the set of all the (?) that makes the corresponding problems have solutions is a relative open set in [0,1] (which will be proved by implicit function theorem), and is also a relative closed set in [0,1] (by a priori estimates), thus the existence of solutions to the problem follows.Since we found that some authors pay more attention to the structure conditions but pay less attention to the "compatibility conditions" which are very important to the parabolic type equations, so we give a little more discussion on this problem in this paper. The problem occurs in such away, that, by the requirement of doing a priori estimates, one need the solutions to have suitably high order regularities up to the boundary, it terns out that this requires the data of the problem must satisfy compatibility conditions up to correspondingly suitably high order. It should be noted that : this is a common requirement to the all family of the problem! But one can only make the data of the problem in study (i.e. for the problem with (?)= 1) satisfy such a kind of requirement, of course, one can also make the data for (?)= 1 to of the same kind of property by construction, then, the compatibility conditions for the problems corresponding to the rest (?)∈[0,1] can be realized only by means of the compatibility conditions of the problems corresponding to these two special values of (?), as well as by suitably constructing the family of problems with one parameter.After series of attempts we found that, owing the non-linearity of the operator in the equation, in order to construct the family of problems with one parameter by the ordinary method, it is necessary that the solutions for (?) = 0 and (?) = 1 must satisfy the same initial conditions!By the above consideration, we choose the following family of problems with one parameter:whereThe difficult appearing in getting the priori estimates is that the equation in (**) is " non-uniformly parabolic" of for the admissible solution. In order to make the use of the methods for "uniformly parabolic equation" , the key locates completing two things: obtaining the positive lower bound of det1/n (D2u? +σ(x)) and the upper bound of |D2u?|0,QT .The usage of structure conditions (C2)(C2') is to guarantee(* * *) det1/n (D2u? +σ(x))has a positive lower bound.But Ladyzhenkaya and Ivochkina only formulated the conditions , there does not show these how to obtain these conditions; In this paper we go a little further to give an approach for deriving these structure conditions: choosing a suitable auxiliary function, by means of comparison theorem, one can derive the structure conditions to ensure the validity of (***). This approach offers a procedure ,by which one not only can re-derive the structure conditions in the works studied by other authors, but also can derive new structure conditions (other than (C2)(C2')) that makes (***) hold by choosing new auxiliary functions.And in estimating the upper bound of |D2u?|0,QT, owing to the difficulty caused by theα(x) in the boundary condition and theσ(x) in the equation, the method used by Ladyzhenkaya and Ivochkina does not apply here; To overcome them, what we do here is to use the trick used by Zhao shengmin and Wang guanglie , as well as the technique used by Ren changyu and Wang guanglie for treatingσ(x) etc .After we obtain these key estimates, it is not hard to prove that, for the solutions in studying, the equation in (**) is an uniformly parabolic one, thus the methods for "uniformly parabolic equation" can be used to get the needed a priori estimates, and then the existence formulated in the theorem follows from the method of continuity.
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