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

In this paper, we are interested in finding the function u(t,x),(t,x)∈[0,T)×Ω from the final data u(T,x)=ϕ(x), satisfies a nonhomogeneous fractional pseudo-parabolic equation. The problem is stable for the cases σ<ν in the sense that the solution of the problem is regularity-loss and we discuss the regularity of the solution to this problem. For the case σ>ν, the problem is ill-posed (in the sense of Hadamard). We propose the general filtering method to regularize this problem. The well-posedness of the regularized problem and some regularity estimates of the regularized solution are obtained. Moreover, error estimates are established under some a priori conditions of the sought solution. The numerical illustrations are given to show the convergence of our method.