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Large-Time Behavior of Solutions to Hyperbolic-Elliptic Coupled Systems

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Abstract.

We are concerned with the asymptotic behavior of a solution to the initial value problem for a system of hyperbolic conservation laws coupled with elliptic equations. This kind of problem was first considered in our previous paper. In the present paper, we generalize the previous results to a broad class of hyperbolic-elliptic coupled systems. Assuming the existence of the entropy function and the stability condition, we prove the global existence and the asymptotic decay of the solution for small initial data in a suitable Sobolev space. Then, it is shown that the solution is well approximated, for large time, by a solution to the corresponding hyperbolic-parabolic coupled system. The first result is proved by deriving a priori estimates through the standard energy method. The spectral analysis with the aid of the a priori estimate gives the second result.

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Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, Fukuoka, 812-8581, Japan

    Shuichi Kawashima & Yoshiko Nikkuni

  2. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, 152-8552, Japan

    Shinya Nishibata

Authors
  1. Shuichi Kawashima
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  2. Yoshiko Nikkuni
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  3. Shinya Nishibata
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Communicated by T.-P. Liu

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Kawashima, S., Nikkuni, Y. & Nishibata, S. Large-Time Behavior of Solutions to Hyperbolic-Elliptic Coupled Systems. Arch. Rational Mech. Anal. 170, 297–329 (2003). https://doi.org/10.1007/s00205-003-0273-6

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  • DOI: https://doi.org/10.1007/s00205-003-0273-6

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