ASYMPTOTIC STABILITY OF SOLUTIONS TO A HYPERBOLIC-ELLIPTIC COUPLED SYSTEM OF RADIATING GAS ON THE HALF LINE

Authors：Ji, Shanming; Zhang, Minyi; Zhu, Changjiang^*

Source：SIAM Journal on Mathematical Analysis, 2022, 54(3): 3884-3929.

DOI：10.1137/21M1459848

• hyperbolic-elliptic coupled system
• initial boundary value problem
• asymptotic behavior
• L-2-energy method
• singular phase plane analysis

This paper is concerned with the asymptotic stability of the solution to an initial-boundary value problem on the half line for a hyperbolic-elliptic coupled system of radiating gas, where the data on the boundary and at the far field state are defined as u(-) and u(+) satisfying u(-) < u(+). For the scalar viscous conservation law case, it is known by the work of Liu, Matsumura, and Nishihara [SIAM J. Math. Anal., 29 (1998), pp. 293-308] that the solution tends toward a rarefaction wave or stationary solution or superposition of these two kinds of waves depending on the distribution of u(+/-). Motivated by their work, we prove the stability of the above three types of wave patterns for the hyperbolic-elliptic coupled system of radiating gas with small perturbation. A singular phase plane analysis method is introduced to show the existence and the precise asymptotic behavior of the stationary solution, especially for the degenerate case: u(-) < u(+) = 0 such that the system has inevitable singularities. The stability of the rarefaction wave, the stationary solution, and their superposition is proved by applying the standard L-2-energy method.