  In this paper stability and error estimates for time discretizations of
linear and semilinear parabolic equations by the two-step backward
differentiation formula (BDF2) method with variable step-sizes are derived. An
affirmative answer is provided to the question: whether the upper bound of
step-size ratios for the $l^\infty(0,T;H)$-stability of the BDF2 method for
linear and semilinear parabolic equations is identical with the upper bound for
the zero-stability. The $l^\infty(0,T;V)$-stability of the variable step-size
BDF2 method is also established under more relaxed condition on the ratios of
consecutive step-sizes. Based on these stability results, error estimates in
several different norms are derived. To utilize the BDF method the trapezoidal
method and the backward Euler scheme are employed to compute the starting
value. For the latter choice, order reduction phenomenon of the constant
step-size BDF2 method is observed theoretically and numerically in several
norms. Numerical results also illustrate the effectiveness of the proposed
method for linear and semilinear parabolic equations.
