We consider quasilinear partially dissipative hyperbolic systems with time-dependent damping in the whole space $\mathbb{R}^d$, with $d \ge 1$. Using an approach similar to that developed by Crin-Barat and Danchin, we establish the global existence of small-amplitude solutions for systems endowed with a damping term of the form $-\frac{K z}{(1+t)^{\alpha}}$, with $0<\alpha\le 1$. We assume that the linearized system satisfies the Shizuta–Kawashima (SK) condition, which ensures that the dissipation acts on all characteristic components through coupling. The key idea is to construct a Lyapunov-type functional that compensates for the lack of full dissipation. Such a functional was first introduced by Beauchard and Zuazua in the framework of control theory.
