Consistent two-sided estimates for the solutions of quasilinear parabolic equations and their approximations

For a linearized finite-difference scheme approximating the Dirichlet problem for a multidimensional quasilinear parabolic equation with unbounded nonlinearity, we establish pointwise two-sided solution estimates consistent with similar estimates for the differential problem. These estimates are used to prove the convergence of finite-difference schemes in the grid L2 norm.