On a final value problem for a nonlinear fractional pseudo-parabolic equation

In this paper, we investigate a final boundary value problem for a class of fractional with parameter β pseudo-parabolic partial differential equations with nonlinear reaction term. For 0<β < 1, the solution is regularity-loss, we establish the well-posedness of solutions. In the case that β >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 modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.