Summary: One parabolic \(p\)-Laplacian-like differential equation with mixed boundaries is studied in this paper, where the item \(\frac{\partial u}{\partial t}\) in the corresponding studies is replaced by \(\alpha\frac{\partial u}{\partial t}\), which makes it more general. The sufficient condition of the existence and uniqueness of non-trivial solution in \(L^2(0,T; L^2(\Omega))\) is presented by employing the techniques of splitting the boundary problems into operator equation. Compared to the corresponding work, the restrictions imposed on the equation are weaken and the proof technique is simplified. It can be regarded as the extension and complement of the previous work.