Decay property of regularity-loss type and nonlinear effects for some hyperbolic-elliptic system

Takekiyo Kubo, Shuichi Kawashima

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

We study the decay property of a certain nonlinear hyperbolic-elliptic system with 2mth-order elliptic part, which is a modified version of the simplest radiating gas model. It is proved that, for m ≥ 2, the system verifies a decay property of the regularity-loss type that is characterized by the parameter m. This dissipative property is very weak in the high-frequency region and causes a difficulty in showing the global existence of solutions to the nonlinear problem. By employing the time-weighted energy method together with the optimal decay for lower-order derivatives of solutions, we overcome this difficulty and establish a global existence and asymptotic decay result. Furthermore, we show that the solution approaches the nonlinear diffusion wave described by the self-similar solution of the Burgers equation as time tends to infinity.

Original languageEnglish
Pages (from-to)139-159
Number of pages21
JournalKyushu Journal of Mathematics
Volume63
Issue number1
DOIs
Publication statusPublished - Jun 19 2009

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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