The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions
The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions
复制标题
DOI：
10.1016/j.jde.2011.03.024
复制
发表时间：
2010-09
期刊：
影响因子：
--
通讯作者：
H. Takamura;Kyouhei Wakasa
中科院分区：
文献类型：
--
作者：
H. Takamura;Kyouhei Wakasa
施特劳斯半线性波动方程猜想的最后一个开放部分是高维临界情况的爆炸定理。 Yordanov和Zhang（2006）[18]或Zhou（2007）[21]独立解决了这个问题。但两篇论文都没有阐明对解的寿命（即最大存在时间）的估计。在本文中，我们改进了他们的定理并引入了新的迭代论证来获得寿命的尖锐上限。因此，根据 Li 和 Zhou (1995)[10] 的尖锐下界，R 4×[0,∞) 中 u t t− Δ u= u 2 的解的寿命 T (ε) 的估计值是 exp （c ε− 2)⩽ T (ε)⩽ exp (C ε− 2)，其中 c 和 C 是仅取决于 f 和 g 的正常数。众所周知，这个上限是完全非线性波动方程一般理论的最后一个开放最优性。
The final open part of Straussʼ 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 t t− Δ u= u 2 in R 4×[0,∞) with the initial data u (x, 0)= ε f (x), u t (x, 0)= ε g (x) of a small parameter ε> 0, compactly supported smooth functions f and g, has an estimate exp (c ε− 2)⩽ T (ε)⩽ exp (C ε− 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.