Global solvability and asymptotic behavior of a free boundary problem for the one-dimensional viscous radiative and reactive gas

Abstract

In this paper, we study a Neumann and free boundary problem for the one-dimensional viscous radiative and reactive gas. We prove that under rather general assumptions on the heat conductivity κ, for any arbitrary large smooth initial data, the problem admits a unique global classical solution. Our global existence results improve those results by Umehara and Tani [“Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas,” J. Differ. Equations 234(2), 439–463 (2007)10.1016/j.jde.2006.09.023; Umehara and Tani “Global solvability of the free-boundary problem for one-dimensional motion of a self-gravitating viscous radiative and reactive gas,” Proc. Jpn. Acad., Ser. A: Math. Sci. 84(7), 123–128 (2008)]10.3792/pjaa.84.123 and by Qin, Hu, and Wang [“Global smooth solutions for the compressible viscous and heat-conductive gas,” Q. Appl. Math. 69(3), 509–528 (2011)].10.1090/S0033-569X-2011-01218-0 Moreover, we analyze the asymptotic behavior of the global solutions to our problem, and we prove that the global solution will converge to an equilibrium as time goes to infinity. This is the result obtained for this problem in the literature for the first time.