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The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions
| Title: | The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions |
| Authors: | Takamura, Hiroyuki Browse this author | | Wakasa, Kyouhei Browse this author |
| Issue Date: | 24-Sep-2010 |
| Publisher: | Department of Mathematics, Hokkaido University |
| Journal Title: | Hokkaido University Preprint Series in Mathematics |
| Volume: | 969 |
| Start Page: | 1 |
| End Page: | 18 |
| Abstract: | The final open part of Strauss’ conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang [17], or Zhou [20] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou [9], the lifespan T(ε) of solutions of utt − Δu = u2 in R4 × [0,∞) with the initial data u(x, 0) = εf(x), ut(x, 0) = εg(x) of a small parameter ε > 0, compactly supported smooth functions f and g, has an estimate exp ( cε −2) ≤ T(ε) ≤ exp ( Cε −2) , where c and C are positive constants depending only on f and g. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations. |
| Type: | bulletin (article) |
| URI: | http://hdl.handle.net/2115/69776 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics
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Submitter: 数学紀要登録作業用
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