The asymptotic behavior to diffusion waves for p-system with space-dependent damping on \(\mathbb{R}_+\) (Q6587539)

The paper deals with initial-boundary value problems for \(2\times 2\)-quasilinear dissipative strictly hyperbolic systems in the domain \(\mathbb{R}_+\times\mathbb{R}_+\), with focus on the so-called \(p\)-system of the form\N\[\Nv_t-u_x=0,\quad u_t+p(v)_x=-\alpha(x)u.\N\]\NThe system is subjected either to Dirichlet or to Neumann boundary conditions on \(x=0\). The paper extends the results obtained by \textit{A. Matsumura} and \textit{K. Nishihara} [SIAM J. Math. Anal. 56, No. 1, 993--1015 (2024; Zbl 1532.35068)] and by \textit{K. Nishihara} and \textit{T. Yang} [J. Differ. Equations 156, No. 2, 439--458 (1999; Zbl 0933.35121)] to the case of initial-boundary value problems with space-dependent damping coefficient \(\alpha(x)\). Under the condition that the initial data are smooth and sufficiently small, the global existence of small smooth solutions is proved by using the time-weighted energy method. Moreover, the asymptotic behavior as \(t\to\infty\) is studied using the Green's function approach. It is shown that the solutions converge to diffusion waves with a polynomial convergence rate.