Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation

• 成果类型: Article
• 署名作者: Kenig, Carlos E.; Merle, Frank
• 署名单位: University of Chicago; CY Cergy Paris Universite
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-008-0031-6
• 发表日期: 2008
• 页码: 147-212
• 关键词: korteweg-devries equation; schrodinger-equation; hyperbolic equations; unique continuation; heat-equations; REGULARITY; compactness; instability; EXISTENCE; PRINCIPLE

摘要

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 solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H 1 than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.