We investigate complete linear Weingarten (LW) spacelike submanifolds immersed with parallel normalized mean curvature vector field and second fundamental form locally timelike in the \((n+p)\) -dimensional semi-Riemannian space form \(\mathbb {N}_{q}^{n+p}(c)\) of index \(1\le q\le p\) with constant curvature c. Under suitable constraints, we establish a sharp integral inequality and we use it to characterize compact (without boundary) totally umbilical submanifolds of \(\mathbb {N}_{q}^{n+p}(c)\) . Afterwards, dealing with a parabolicity criterion and an assumption that the gradient of the mean curvature function has Lebesgue integrable norm, we show that a complete LW spacelike submanifold must be either totally umbilical or a product \(M_1\times M_2\times \cdots \times M_k\) , where the factors \(M_i\) are mutually perpendicular along their intersections and totally umbilical submanifolds of \(\mathbb {N}^{n+p}_q(c)\) . Furthermore, we apply a maximum principle related to convergence to zero at infinity in order to prove that a complete noncompact LW spacelike submanifold of \(\mathbb {N}_{q}^{n+p}(c)\) must be totally umbilical.
