Global asymptotics for the damped wave equation with absorption in higher dimensional space

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We consider the Cauchy problem for the damped wave equation with absorption<br><i>u</i><sub><i>tt</i></sub>-Δ<i>u</i>+<i>u</i><sub>t</sub>+|<i>u</i>|<sup>ρ-1</sup><i>u</i> = 0,   (<i>t,x</i>)∈<b><i>R</i></b><sub>+</sub>×<b><i>R</i></b><sup><i>N</i></sup>,   (*)<br>with <i>N</i>=3,4. The behavior of <i>u</i> as <i>t</i>→∞ is expected to be the Gauss kernel in the supercritical case ρ>ρ<sub><i>c</i></sub>(<i>N</i>):=1+2/<i>N</i>. In fact, this has been shown by Karch [<b>12</b>] (Studia Math., 143 (2000), 175-197) for ρ>1+$¥frac{4}{N}$ (<i>N</i>=1,2,3), Hayashi, Kaikina and Naumkin [<b>8</b>] (preprint (2004)) for ρ>ρ<sub><i>c</i></sub>(<i>N</i>) (<i>N</i>=1) and by Ikehata, Nishihara and Zhao [<b>11</b>] (J. Math. Anal. Appl., <b>313</b> (2006), 598-610) for ρ<sub><i>c</i></sub>(<i>N</i>)<ρ≤1+$¥frac{4}{N}$ (<i>N</i>=1,2) and ρ<sub><i>c</i></sub>(<i>N</i>)<ρ<1+$¥frac{3}{N}$ (<i>N</i>=3). Developing their result, we will show the behavior of solutions for ρ<sub><i>c</i></sub>(<i>N</i>)<ρ≤1+$¥frac{4}{N}$ (<i>N</i>=3), ρ<sub><i>c</i></sub>(<i>N</i>)<ρ<1+$¥frac{4}{N}$ (<i>N</i>=4). For the proof, both the weighted <i>L</i><sup>2</sup>-energy method with an improved weight developed in Todorova and Yordanov [<b>22</b>] (J. Differential Equations, <b>174</b> (2001), 464-489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for <i>N</i>=5, because the semilinear term is not in <i>C</i><sup>2</sup> and the second derivatives are necessary when the explicit formula of solutions is estimated.

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  • Journal of the Mathematical Society of Japan  

    Journal of the Mathematical Society of Japan 58(3), 805-836, 2006-07-01 

    The Mathematical Society of Japan

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各種コード

  • NII論文ID(NAID)
    10018381129
  • NII書誌ID(NCID)
    AA0070177X
  • 本文言語コード
    ENG
  • 資料種別
    ART
  • ISSN
    00255645
  • NDL 記事登録ID
    7987144
  • NDL 雑誌分類
    ZM31(科学技術--数学)
  • NDL 請求記号
    Z53-A209
  • データ提供元
    CJP書誌  CJP引用  NDL  J-STAGE 
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