Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation
Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation
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DOI：
10.1007/s11511-008-0031-6
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发表时间：
2006-10
期刊：
影响因子：
3.7
通讯作者：
C. Kenig;F. Merle
中科院分区：
文献类型：
--
作者：
C. Kenig;F. Merle
我们使用能量空间中的 3、4 和 5 维数据研究了能量临界聚焦非线性波动方程。我们证明，对于能量小于给出 Sobolev 嵌入中最佳常数的静态解 W 的柯西数据，以下替代方案成立。如果初始数据在齐次 Sobolev 空间 H1 中的范数小于 W 中的范数，则我们具有全局适定性和散射。如果范数大于W，那么我们就会在有限时间内崩溃。
We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solutionWwhich gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev spaceH1than the one ofW, then we have global well-posedness and scattering. If the norm is larger than the one ofW, then we have break-down in finite time.