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

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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## Detail(s)

Original language English 439-458 Journal of Differential Equations 156 2 Published - 10 Aug 1999

## Abstract

We consider the asymptotic behaviour of solutions to the p-system with linear damping on the half-line R+=(0, ∞), vt-ux=0,ut+p(v)x=-αu, with the Dirichlet boundary condition u single rule fence sign ;x=0=0 or the Neumann boundary condition uxsingle 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+. © 1999 Academic Press.

## Citation Format(s)

[ RIS ] [ BibTeX ]
In: Journal of Differential Equations, Vol. 156, No. 2, 10.08.1999, p. 439-458.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review