Summary: We study a Cauchy problem for a parabolic equation in multiple dimensions, which is naturally a generalization of some one-dimensional and two-dimensional inverse heat conduction problems. This is a severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data. After simply analyzing the ill-posedness of the Cauchy problem in the frequency space, from a new viewpoint we propose two regularization methods: Tikhonov method and Fourier truncation method. We give and prove the convergence estimate between the exact solution and its regularized approximation. We also discuss the relationship of these two and other regularization methods. At last, we employ some numerical examples to illustrate the behavior of the proposed methods.