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

• Nishihara Kenji

  School of Political Science and Economics, Waseda Unviversity

抄録

We consider the Cauchy problem for the damped wave equation with absorption<br>u_tt-Δu+u_t+|u|^ρ-1u = 0,   (t,x)∈R₊×R^N,   (*)<br>with N=3,4. The behavior of u as t→∞ is expected to be the Gauss kernel in the supercritical case ρ>ρ_c(N):=1+2/N. In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175-197) for ρ>1+$¥frac{4}{N}$ (N=1,2,3), Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for ρ>ρ_c(N) (N=1) and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598-610) for ρ_c(N)<ρ≤1+$¥frac{4}{N}$ (N=1,2) and ρ_c(N)<ρ<1+$¥frac{3}{N}$ (N=3). Developing their result, we will show the behavior of solutions for ρ_c(N)<ρ≤1+$¥frac{4}{N}$ (N=3), ρ_c(N)<ρ<1+$¥frac{4}{N}$ (N=4). For the proof, both the weighted L²-energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464-489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for N=5, because the semilinear term is not in C² and the second derivatives are necessary when the explicit formula of solutions is estimated.

収録刊行物

• Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ

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

  一般社団法人 日本数学会

キーワード

• semilinear damped wave equation
• critical exponent
• global asymptotics
• weighted energy method
• explicit formula

詳細情報

• CRID: 1390282680093317760
• NII論文ID: 10018381129
• NII書誌ID: AA0070177X
• DOI: 10.2969/jmsj/1156342039
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• MRID: 2254412
• NDL書誌ID: 7987144
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I7987144
• 本文言語コード: en
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