Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system

Takafumi Hosono, Shuichi Kawashima

    Research output: Contribution to journalArticlepeer-review

    74 Citations (Scopus)

    Abstract

    We discuss the global solvability and asymptotic behavior of solutions to the Cauchy problem for some nonlinear hyperbolic-elliptic system with a fourth-order elliptic part. This system is a modified version of the simplest radiating gas model and verifies a decay property of regularity-loss type. Such a dissipative structure also appears in the dissipative Timoshenko system studied by Rivera and Racke. This dissipative property is very weak in high frequency region and causes the difficulty in deriving the desired a priori estimates for global solutions to the nonlinear problem. In fact, it turns out that the usual energy method does not work well. We overcome this difficulty by employing a time-weighted energy method which is combined with the optimal decay for lower order derivatives of solutions, and we establish a global existence and asymptotic decay result. Furthermore, we show that the solution has an asymptotic self-similar profile described by the Burgers equation as time tends to infinity.

    Original languageEnglish
    Pages (from-to)1839-1859
    Number of pages21
    JournalMathematical Models and Methods in Applied Sciences
    Volume16
    Issue number11
    DOIs
    Publication statusPublished - Nov 2006

    All Science Journal Classification (ASJC) codes

    • Modelling and Simulation
    • Applied Mathematics

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