Uniqueness of the solution of the first mixed problem for a two-dimentional nonlinear parabolic equation

The first mixed problem for two-dimentional nonlinear parabolic equation with nonlinear second derivatives of a desired function is considered. We assume that the solution possessing continuous second derivatives with respect to coordinate variables exists in a closed cylinder under some restrictions on initial data of the problem. The uniqueness of this problem is proved by using the longitudinal version of the method of lines.