$l_k$-biharmonic spacelike hypersurfaces in minkowski $4$-space $mathbb{e}_1^4$

biharmonic surfaces in euclidean space $mathbb{e}^3$ are firstly studied from a differential geometric point of view by bang-yen chen, who showed that the only biharmonic surfaces are minimal ones. a surface $x : m^2rightarrowmathbb{e}^{3}$ is called biharmonic if $delta^2x=0$, where $delta$ is the laplace operator of $m^2$. we study the $l_k$-biharmonic spacelike hypersurfaces in the $4$-dimentional pseudo-euclidean space $mathbb{e}_1^4$ with an additional condition that the principal curvatures of $m^3$ are distinct. a hypersurface $x: m^3rightarrowmathbb{e}^{4}$ is called $l_k$-biharmonic if $l_k^2x=0$ (for $k=0,1,2$), where $l_k$ is the linearized operator associated to the first variation of $(k+1)$-th mean curvature of $m^3$. since $l_0=delta$, the matter of $l_k$-biharmonicity is a natural generalization of biharmonicity. on any $l_k$-biharmonic spacelike hypersurfaces in $mathbb{e}_1^4$ with distinct principal curvatures, by, assuming $h_k$ to be constant, we get that $h_{k+1}$ is constant. furthermore, we show that $l_k$-biharmonic spacelike hypersurfaces in $mathbb{e}_1^4$ with constant $h_k$ are $k$-maximal.