In this paper, we investigate a nal boundary value problem for a class of fractional with parameter pseudo-parabolic partial di erential equa- tions with nonlinear reaction term. For 0 < < 1; the solution is regularity- loss, we establish the well-posedness of solutions. In the case that > 1, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modi ed Lavrentiev regularization method and Fourier truncated regulariza- tion method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established
