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Volume 21, Issue 2
Asymptotic Decay Toward Rarefaction Wave for a Hyperbolic-elliptic Coupled System on Half Space

Lizhi Ruan & Changjiang Zhu

J. Part. Diff. Eq., 21 (2008), pp. 173-192.

Published online: 2008-05

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  • Abstract
We consider the asymptotic behavior of solutions to a model of hyperbolic- elliptic coupled system on the half-line R_+ = (0,∞), u_t+uu_x+q_x=0, -q_{xx}+q+u_x=0, with the Dirichlet boundary condition u(0, t) = 0. S. Kawashima and Y. Tanaka [Kyushu J. Math., 58(2004), 211-250] have shown that the solution to the correspond- ing Cauchy problem behaviors like rarefaction waves and obtained its convergence rate when u_-< u_+. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, asymptotic behavior of the solution (u, q) is proved to be rarefaction wave as t tends to infinity. Its convergence rate is also obtained by the standard L^2-energy method and L^1-estimate. It decays much lower than that of the corresponding Cauchy problem.
  • Keywords

Hyperbolic-elliptic coupled system rarefaction wave asymptotic decay rate half space L^2-energy method L^1-estimate

  • AMS Subject Headings

35B40 35B45 35G30 35M10 35Q35.

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COPYRIGHT: © Global Science Press

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@Article{JPDE-21-173, author = {}, title = {Asymptotic Decay Toward Rarefaction Wave for a Hyperbolic-elliptic Coupled System on Half Space}, journal = {Journal of Partial Differential Equations}, year = {2008}, volume = {21}, number = {2}, pages = {173--192}, abstract = { We consider the asymptotic behavior of solutions to a model of hyperbolic- elliptic coupled system on the half-line R_+ = (0,∞), u_t+uu_x+q_x=0, -q_{xx}+q+u_x=0, with the Dirichlet boundary condition u(0, t) = 0. S. Kawashima and Y. Tanaka [Kyushu J. Math., 58(2004), 211-250] have shown that the solution to the correspond- ing Cauchy problem behaviors like rarefaction waves and obtained its convergence rate when u_-< u_+. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, asymptotic behavior of the solution (u, q) is proved to be rarefaction wave as t tends to infinity. Its convergence rate is also obtained by the standard L^2-energy method and L^1-estimate. It decays much lower than that of the corresponding Cauchy problem.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5276.html} }
TY - JOUR T1 - Asymptotic Decay Toward Rarefaction Wave for a Hyperbolic-elliptic Coupled System on Half Space JO - Journal of Partial Differential Equations VL - 2 SP - 173 EP - 192 PY - 2008 DA - 2008/05 SN - 21 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5276.html KW - Hyperbolic-elliptic coupled system KW - rarefaction wave KW - asymptotic decay rate KW - half space KW - L^2-energy method KW - L^1-estimate AB - We consider the asymptotic behavior of solutions to a model of hyperbolic- elliptic coupled system on the half-line R_+ = (0,∞), u_t+uu_x+q_x=0, -q_{xx}+q+u_x=0, with the Dirichlet boundary condition u(0, t) = 0. S. Kawashima and Y. Tanaka [Kyushu J. Math., 58(2004), 211-250] have shown that the solution to the correspond- ing Cauchy problem behaviors like rarefaction waves and obtained its convergence rate when u_-< u_+. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, asymptotic behavior of the solution (u, q) is proved to be rarefaction wave as t tends to infinity. Its convergence rate is also obtained by the standard L^2-energy method and L^1-estimate. It decays much lower than that of the corresponding Cauchy problem.
Lizhi Ruan & Changjiang Zhu . (2019). Asymptotic Decay Toward Rarefaction Wave for a Hyperbolic-elliptic Coupled System on Half Space. Journal of Partial Differential Equations. 21 (2). 173-192. doi:
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