Long-time behavior of solutions to nonlinear evolution equations
Long-time behavior of solutions to nonlinear evolution equations
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DOI：
10.1007/bf00253225
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发表时间：
1982-03
影响因子：
2.5
通讯作者：
S. Klainerman
中科院分区：
文献类型：
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作者：
S. Klainerman
我们对非线性演化偏微分方程解的大时间行为感兴趣。更准确地说，我们处理非线性方程的初值问题，这些方程是经典线性方程的扰动，例如：波动方程、克莱因-戈登方程、线性各向同性弹性方程、热方程、薛定谔方程等。目的是表明，对于足够小的非线性扰动，相应的解与线性方程的解一样渐近。特别是，我们恢复并扩展了我们早期关于非线性波动方程全局存在性的结果[1]。正如在那项工作中一样，我们的方法依赖于强大的 Nash-Moser-H6rmander 方案的特殊版本，该方案使我们能够处理非常大类的扰动。该方法似乎特别适合双曲方程；然而，我们在这里尝试处理其他经典方程，例如热方程和薛定谔方程。对于一些具体的例子，我们的方法过于粗糙和不加区别，无法给出最佳结果。特别是，对于特殊类别的半线性扰动，我们的结果弱于文献中的结果。然而，我们可以允许这里的扰动通常包含与线性部分相同阶的导数，对于半线性情况中使用的经典方法似乎不起作用。
We are interested in large time behavior of solutions to nonlinear partial differential equations of evolution. More precisely we deal with the Initial-Value Problem for nonlinear equations which are perturbations of classical linear equations like: the Wave Equation, Klein-Gordon Equation, Linear Isotropic Elasticity Equations, Heat Equation, Schr6dinger Equation, etc. The aim is to show that for sufficiently small nonlinear perturbations the corresponding solutions behave asymptotically like the solutions of the linear equations. In particular, we recover and extend our earlier results on global existence for nonlinear wave equations [1]. As in that work our method relies on a special version of the powerful Nash-Moser-H6rmander scheme which allows us to treat a very large class of perturbations. The method seems to be particularly suited for hyperbolic equations; however, we attempt here a treatement of other classical equations like the Heat Equation and the Schr6dinger Equation. For some specific examples our method is too rough and indiscriminate to give optimal results. In particular, for the special class of semilinear perturbations our results are weaker than what is available in the literature. However, we can allow here perturbations which contain, in general, derivatives of the same order as the linear part, for which the classical methods used in the semilinear case do not seem to work.