In this work, we focus on the final value problem of an inverse problem for both linear and nonlinear biharmonic equations. The aim of this study is to provide a regularized method for the bi-harmonic equation, once the observed data are obtained at a terminal time in $L^q(\mathrm{\Omega })$. We obtain an approximated solution using the Fourier series truncation method and the terminal input data in $L^q(\mathrm{\Omega })$ for $q\ne 2$. In comparision with previous studies, the most highlight of this study is the error between the exact and regularized solutions to be estimated in $L^q(\mathrm{\Omega })$; wherein an embedding between $L^q(\mathrm{\Omega })$ and Hilbert scale spaces ${\mathcal{H}}^{\rho }(\mathrm{\Omega })$ is applied.