Decay of a model system of radiating gas

Abstract

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 $p\in(1,2]$.Comment: 15 page

Asymptotic stability of solutions to a hyperbolic-elliptic coupled system of the radiating gas on the half line

“…Ruan and Zhang in [31] further studied the case: u − < u + for general flux f (u). The radiating gas system (1.4) in the following scalar equation form with convolution u t + f (u) x + u − ψ * u = 0, x ∈ R, t > 0, (1.5) where ψ(x) is the fundamental solution to the elliptic operator −∆ + I in R, was studied in [2,4,17,21,28,35,38]. It was Schochet and Tadmor in [34] who first proved the W 1,∞ regularity of the solution to (1.5).…”