In this paper, we prove the existence of the weak solution for a second-order parabolic equation with a nonlinear diffusion term and present the finite difference method to solve it. We prove the existence of the approximate solution and the generalized solution.Then we provide the difference scheme to approximate the temporal derivatives and spatial derivative. The numerical results show that the proposed difference scheme is simple, easy to implement with high accuracy. In addition, the implicit scheme could be employed while the long-time behavior is considered.