Global solution to the two-dimensional Klein-Gordon equation
Global solution to the two-dimensional Klein-Gordon equation
复制标题
二维 Klein-Gordon 方程的全局解
DOI：
10.1080/03605309108820786
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发表时间：
1991
影响因子：
1.9
通讯作者：
P. Popivanov
中科院分区：
文献类型：
--
作者：
V. Georgiev;P. Popivanov
在波动方程的情况下，[111，[151，[171]]中研究了类似的情况。事实上，如果nr4.0，则保证了非线性波动方程小振幅解的存在性。对于 n53，只能估计寿命，因为 F. John [131、B. Hanouzet 和 J.-L. 构建了反例。 Joly [91] 表明解在有限时间内爆炸。另一方面，在[I71中，Klainerman引入了二次非线性的代数条件，该条件产生了临界情况n= 3的非线性波动方程的全局解的存在性。[111,[121,[161]中的结果表明，n=2的情况对于非线性Klein-Gordon方程是关键的。这项工作的主要目标是为二维 Klein-Gordon 方程，以便只要初始数据是小 C: 函数，就存在全局解。更准确地说，我们研究柯西问题
In case of wave equation a similar situation was studied in [111,[151,[171. In fact, the existence of small amplitude solution to the nonlinear wave equation is guaranteed if nr4. For n53 only estimate of the life-span is possible since the counterexamples constructed by F. John [131, B. Hanouzet and J.-L. Joly [91 show that the solution blows-up for a finite time. On the other hand, in [I71 Klainerman introduced an algebraic condition on the quadratic nonlinearity which yields the existence of a global solution to the nonlinear wave equation for the critical case n= 3. The results in [111,[121,[161 show that the case n= 2 is critical for the nonlinear Klein-Gordon equation.The main goal of this work is to introduce a suitable algebraic condition on the quadratic nonlinearity for the two-dimensional Klein-Gordon equation so that a global solution exists whenever the initial data are small C: functions. More precisely, we study the Cauchy problem