Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping

We consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R₊=(0, ∞), v_t-u_x=0,u_t+p(v)_x=-αu, with the Dirichlet boundary condition u single rule fence sign ;_x=0=0 or the Neumann boundary condition u_xsingle rule fence sign _x=0=0. The initial date (v₀, u₀)(x) has the constant state (v₊, u₊) at x=∞. L. Hsiao and T.-P. Liu [Commun. Math. Phys.143 (1992), 599-605] have shown that the solution to the corresponding Cauchy problem behaves like diffusion wave, and K. Nishihara [J. Differential Equations131 (1996), 171-188; 137 (1997), 384-395] has proved its optimal convergence rate. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, the solution (v, u) is proved to tend to (v₊, 0) as t tends to infinity. Its optimal convergence rate is also obtained by using the Green function of the diffusion equation with constant coefficients. In the case of null-Neumann boundary condition on u, v(0, t) is conservative and v(0, t)≡v₀(0) by virtue of the first equation, so that v(x, t) is expected to tend to the diffusion wave v(x, t) connecting v₀(0) and v₊. In fact the solution (v, u)(x, t) is proved to tend to (v(x, t), 0). In the special case v₀(0)=v₊, the optimal convergence rate is also obtained. However, this is not known in the case v₀(0)≠v₊.