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

Kenji Nishihara, Tong Yang

    Research output: Contribution to journalArticle

    58 Citations (Scopus)

    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+.

    Original languageEnglish
    Pages (from-to)439-458
    Number of pages20
    JournalJournal of Differential Equations
    Volume156
    Issue number2
    Publication statusPublished - 1999 Aug 10

    Fingerprint

    Boundary Effect
    Asymptotic Behavior of Solutions
    Optimal Convergence Rate
    Damping
    Boundary conditions
    Tend
    Fences
    Neumann Boundary Conditions
    Dirichlet Boundary Conditions
    Null
    Date
    Green's function
    Diffusion equation
    Half line
    Cauchy Problem
    Infinity
    Coefficient

    ASJC Scopus subject areas

    • Analysis

    Cite this

    Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping. / Nishihara, Kenji; Yang, Tong.

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

    Research output: Contribution to journalArticle

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