  In this paper we consider a linearized variable-time-step two-step backward
differentiation formula (BDF2) scheme for solving nonlinear parabolic
equations. The scheme is constructed by using the variable time-step BDF2 for
the linear term and a Newton linearized method for the nonlinear term in time
combining with a Galerkin finite element method (FEM) in space. We prove the
unconditionally optimal error estimate of the proposed scheme under mild
restrictions on the ratio of adjacent time-steps, i.e. $0<r_k < r_{\max}
\approx 4.8645$ and on the maximum time step. The proof involves the discrete
orthogonal convolution (DOC) and discrete complementary convolution (DCC)
kernels, and the error splitting approach. In addition, our analysis also shows
that the first level solution $u^1$ obtained by BDF1 (i.e. backward Euler
scheme) does not cause the loss of global accuracy of second order. Numerical
examples are provided to demonstrate our theoretical results.
