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We give a new decay framework for general dissipative hyperbolic system andhyperbolic-parabolic composite system, which allow us to pay less attention onthe traditional spectral analysis in comparison with previous efforts. Newingredients lie in the high-frequency and low-frequency decomposition of apseudo-differential operator and an interpolation inequality related tohomogeneous Besov spaces of negative order. Furthermore, we develop theLittlewood-Paley pointwise energy estimates and new time-weighted energyfunctionals to establish the optimal decay estimates on the framework ofspatially critical Besov spaces for degenerately dissipative hyperbolic systemof balance laws. Based on the $L^{p}(\mathbb{R}^{n})$ embedding and improvedGagliardo-Nirenberg inequality, the optimal$L^{p}(\mathbb{R}^{n})$-$L^{2}(\mathbb{R}^{n})(1\leq p<2)$ decay rates and$L^{p}(\mathbb{R}^{n})$-$L^{q}(\mathbb{R}^{n})(1\leq p<2\leq q\leq\infty)$decay rates are further shown. Finally, as a direct application, the optimaldecay rates for 3D damped compressible Euler equations are also obtained.

The optimal decay estimates on the framework of Besov spaces for generally dissipative systems