In the present paper, a priori estimates of the stability in the sense of the initial data of the difference schemes approximating quasilinear parabolic equations and nonlinear transfer equation have been obtained. The basic point is connected with the necessity of estimating all derivatives entering into the nonlinear part of the difference equations. These estimates have been proved without any assumptions about the properties of the differential equations and depend only on the behavior of the initial and boundary conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for the finite instant of time t 6 t0 connected with the fact that the solution of the Riccati equation becomes infinite. is already associated with the behavior of the second derivative of the initial function and coincides with the time of the exact solution destruction (heat localization in the peaking regime). A close relation between the stability and convergence of the difference scheme solution is given. Thus, not only a priori estimates for stability have been established, but it is also shown that the obtained conditions permit exact determination of the time of destruction of the solution of the initial boundary value problem for the original nonlinear differential equation in partial derivatives. In the present paper, concrete examples confirming the theoretical conclusions are given.