The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

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• 作者：Hiroyuki Takamura^a ; ^{takamura@fun.ac.jp"" rel=""nofollow} ; Kyouhei Wakasa^b ; ¹ ; ^{g2111045@fun.ac.jp"" rel=""nofollow}
• 关键词：Lifespan ; Semilinear wave equation ; Critical exponent ; High dimensions
• 刊名：Journal of Differential Equations
• 出版年：2011
• 出版时间：15 August 2011
• 年：2011
• 卷：251
• 期：4-5
• 页码：1157-1171
• 全文大小：195 K

文摘

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 (2006) [18], or Zhou (2007) [21] 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 (1995) [10], the lifespan T(ε) of solutions of u_tt−Δu=u² in R⁴×[0,∞) with the initial data u(x,0)=εf(x),u_t(x,0)=εg(x) of a small parameter ε>0, compactly supported smooth functions f and g, has an estimateexp(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.