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

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 . del u(2) + divq = 0, -del divq + q + del u = 0, with initial data u(x(t),..., x(n), 0) = u(0)(x(1),..., x(n)) -> U+/-, x(1) -> +/-infinity. 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. (C) 2010 Elsevier Inc. All rights reserved.