Long-time dynamics of partially dissipative hyperbolic systems with non-autonomous coefficients

RESUMO

We study quasilinear symmetrizable partially dissipative hyperbolic systems with non-autonomous relaxation coefficients in $\mathbb{R}^d$ ($d\geq1$). The existence of global strong solutions is established in a critical regularity setting for systems satisfying the so-called Shizuta-Kawashima (SK) and entropy conditions. When the initial data are additionally bounded in a lower-regularity norm, we prove that the corresponding solutions converge to equilibrium at optimal algebraic decay rates. Furthermore, we show that the conservative part of the solution behaves asymptotically as the solution of a non-autonomous parabolic equation. Our results apply to the compressible Euler system with the time-dependent damping coefficient $\frac{K}{(1+t)^{\alpha}}$ ($\alpha<1$, $K>0$ or $\alpha=1$, $K\gg 1$) in the velocity equation. The natural low/high-frequency splitting of the autonomous theory persists in the non-autonomous setting, but with a frequency-threshold that evolves in time. To handle this moving frequency structure, we introduce a new class of hybrid Besov spaces adapted to time-dependent thresholds and derive hypocoercive estimates in each frequency regime. Our results reveal the qualitative and quantitative effects of general time-dependent relaxation coefficients on dissipation and large-time dynamics.