The quasi-reversibility method to solve the Cauchy problems for parabolic equations

In this paper, on the one hand, we take the conventional quasi-reversibility method to obtain the error estimates of approximate solutions of the Cauchy problems for parabolic equations in a sub-domain of Q T with strong restrictions to the measured boundary data. On the other hand, weakening the conditions on the measured data, then combining the duality method in optimization with the quasi-reversibility method, we solve the Cauchy problems for parabolic equations in the presence of noisy data. Using this method, we can get the proper regularization parameter . that we need in the quasi-reversibility method and obtain the convergence rate of approximate solutions as the noise of amplitude . tends to zero.