  This paper is concerned with numerical solutions of time-fractional nonlinear
parabolic problems by a class of $L1$-Galerkin finite element methods. The
analysis of $L1$ methods for time-fractional nonlinear problems is limited
mainly due to the lack of a fundamental Gronwall type inequality. In this
paper, we establish such a fundamental inequality for the $L1$ approximation to
the Caputo fractional derivative. In terms of the Gronwall type inequality, we
provide optimal error estimates of several fully discrete linearized Galerkin
finite element methods for nonlinear problems. The theoretical results are
illustrated by applying our proposed methods to three examples: linear
Fokker-Planck equation, nonlinear Huxley equation and Fisher equation.
