Scattering for the two-dimensional energy-critical wave equation
Scattering for the two-dimensional energy-critical wave equation
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DOI：
10.1215/00127094-2009-053
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发表时间：
2009-11
影响因子：
2.5
通讯作者：
S. Ibrahim;M. Majdoub;N. Masmoudi;K. Nakanishi
中科院分区：
文献类型：
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作者：
S. Ibrahim;M. Majdoub;N. Masmoudi;K. Nakanishi
研究了二维空间中具有散焦指数非线性项的非线性Klein-Gordon和Schr“odinger方程的波算子的存在性和渐近完备性.基于守恒哈密顿量的值定义了一定的阈值，在该阈值以下，指数势能通过Trudinger-Moser型不等式由动能支配。我们证明，如果能量低于或等于临界值，则解在无穷大时间接近自由Klein-Gordon解。在临界情况下的有趣的特点是，与Sobolev型不等式一起的Eschenhartz估计不能控制非线性项均匀的每个时间间隔，但常数取决于有多少解决方案集中。因此，我们必须追踪能量沿着时间的集中，以便建立有利的非线性估计，然后实施布尔甘的归纳论证。对“亚临界”非线性Schr odinger方程也得到了同样的结果.
We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain's induction argument. We show the same result for the "subcritical" nonlinear Schr\"odinger equation.