Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain
Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain
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有界域中能量临界二维波动方程的 Strichartz 型估计和适定性
DOI：
10.1016/j.jde.2011.01.008
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发表时间：
2010
影响因子：
2.4
通讯作者：
Rym Jrad
中科院分区：
文献类型：
--
作者：
Slim Ibrahim;Rym Jrad
研究了光滑有界区域Ω <$R2上H1-临界半线性波动方程带Dirichlet或Neumann边界条件的Cauchy问题的适定性.首先，我们证明了一个适当的Lqspectral投影估计的拉普拉斯运营商的Lqspectral投影型估计。我们的证明遵循Burq，Lebeau和Planchon（2008）[4]。然后，我们证明了当能量低于或等于由尖锐的Moser-Trudinger不等式给出的阈值时的整体适定性。最后，在超临界的情况下，我们证明了不稳定的结果，使用有限的传播速度和相关的ODE振荡数据的定量研究。
We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H1-critical semilinear wave equation on a smooth bounded domain Ω⊂R2. First, we prove an appropriate Strichartz type estimate using the Lqspectral projector estimates of the Laplace operator. Our proof follows Burq, Lebeau and Planchon (2008) [4]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser–Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data.