This paper is concerned with optimal time-decay estimates of solutions of the Cauchy problem to a model system of the radiating gas in $\mathbb{ R}^n$. Compared to Liu and Kawashima (2011) \cite{ Liu1} and Wang and Wang (2009) \cite{ Wang}, without smallness assumption of initial perturbation in $L^1$-norm, we study large time behavior of small amplitude classical solutions to the Cauchy problem. The optimal $H^N$-norm time-decay rates of the solutions in $\mathbb{ R}^n$ with $1\leq n\leq4$ are obtained by applying the Fourier splitting method introduced in Schonbek (1980) \cite{ Schonbek1} with a slight modification and an energy method. Furthermore, basing on a refined pure energy method introduced in Guo and Wang \cite{ Guo} (2011), we give optimal $L^p$-$L^2(\mathbb{ R}^3)$ decay estimates of the derivatives of solutions when initial perturbation is bounded in $L^p$-norm with some $p\in(1,2]$.
Categories: math.AP