Asymptotic behavior of solutions to the system of compressible adiabatic flow through porous media (Q2719188)

The system under consideration reads NEWLINE\[NEWLINE\partial_t v - \partial_x u= 0,\quad \partial_t u + \partial_x p(v,s)= - \alpha u,\quad \partial_t s = 0,NEWLINE\]NEWLINE where the parameter \(\alpha\) is positive, and the pressure law is such that \(p>0\), \(\partial_v p(r,s)<0\) for \(v>0\). The asymptotic behavior of solutions for initial data having limits \((\underline{v},u_\pm,\underline{s})\) at \(\pm\infty\) was first studied by \textit{L. Hsiao (Xiao)} and \textit{D. Serre} [Chin. Ann. Math., Ser. B 16, No. 4, 431-444 (1995; Zbl 0836.35022), SIAM J. Math. Anal. 27, No. 1, 70-77 (1996; Zbl 0849.35069)]. Here a convergence rate is obtained, in continuation of a former work of the first author [J. Differ. Equations 131, No. 2, 171-188 (1996; Zbl 0866.35066)] in the isentropic case (\(s\equiv\) constant). The technique of proof makes use of the Green's function for the parabolic equation NEWLINE\[NEWLINE\partial_t V + \partial_x(\partial_v p(\underline{v},\underline{s}) \partial_x V) = 0.NEWLINE\]