Asymptotic behavior of solutions to a hyperbolic–elliptic coupled system in multi-dimensional radiating gas

Abstract

This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem of a hyperbolic–elliptic coupled system in the multi-dimensional radiating gas u t + a ⋅ ∇ u 2 + div q = 0 , − ∇ div q + q + ∇ u = 0 , with initial data u ( x 1 , … , x n , 0 ) = u 0 ( x 1 , … , x n ) → u ± , x 1 → ± ∞ . First, for the case with the same end states u − = u + = 0 , we prove the existence and uniqueness of the global solutions to the above Cauchy problem by combining some a priori estimates and the local existence based on the continuity argument. Then L p -convergence rates of solutions are respectively obtained by applying L 2 -energy method for n = 1 , 2 , 3 and L p -energy method for 3 < n < 8 and interpolation inequality. Furthermore, by semigroup argument, we obtain the decay rates to the diffusion waves for 1 ⩽ n < 8 . Secondly, for the case with the different end states u − < u + , our main concern is that the corresponding Cauchy problem in n-dimensional space ( n = 1 , 2 , 3 ) behaviors like planar rarefaction waves. Its convergence rate is also obtained by L 2 -energy method and L 1 -estimate.