Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence

where n is an open-bounded subset of IRN, 0 < T < oo and a function F : (t,x,u,p,X) E [O,T] X n X IR X IRN X MN--+ F(t,x,u,p,X) E IRis given. Here MN denotes the space of N x N symmetric matrices, Du = gradxu = (ux,, · · · , UxN ), D 2u is the Hessian matrix of u with respect to x = (x 1 , · · · ,xN)· As it is well known, the notion of viscosity solution was introduced by Crandall and Lions [3] for the Hamilton-Jacobi equations (see also Crandall, Evans and Liops [1] for some equivalent formulations and simplifications of proofs in [3]) and th'en extended by Lions [14] to the second-order case; we also refer to [13], [8], [9] and the references therein as far as the existence and uniqueness of such solutions is concerned. Since the definition of viscosity solution involves pointwise values of F, the functions F considered in [1], [3] and [14] are supposed to be continuous in all arguments. Recently, H. Ishii in [7] has solved the problem of the uniqueness and the existence of solutions of (1.1) where F does not depend on D 2u, is integrable in (0, T) and continuous in the remaining arguments. To do so, he extended the definition of viscosity solution to the case of noncontinuous F, by looking at the pointwise behavior of a class of continuous functions related in some local way to F. Notice that in [15) are presented some equivalent definitions originating from the theory of acretive operators. Here we extend this new definition of viscosity solution to the case ( 1.1) of the second-order equations. We make the following assumptions