论文标题
半线性进化方程的Hartman-Grobman定理中同态的较高规律性
Higher regularity of homeomorphisms in the Hartman-Grobman theorem for semilinear evolution equations
论文作者
论文摘要
Hein和Prüss[J。微分方程，261（2016）4709-4727]呈现了半连续性双曲进化方程的Hartman-Grobman类型$ C^{0} $线性化结果。他们表明，线性化图（同构）及其逆性是Hölder连续的。一个重要的问题：是否有可能提高同态的规律性？在本文中，我们证明，如果半线性系统的轻度溶液有界，那么同态的规律性是Lipchitzian，但逆性仅是Hölder连续的。在本文中，我们还给出了广义的局部线性化结果。最后，某些应用程序结束了论文。正如背面指出的那样[J.微分方程，297（2021）536-574]，即使差异$ f $是$ c^{\ infty} $，同构也可能无法本地lipschitz。同态通常仅在本地连续存在。但是，通过建立两个有效的二分法整体不平等，我们证明了共轭性是lipchitzian，但逆向是hölder连续的。我们的结果是第一个观察Hartman-Grobman定理中同态较高规律性的结果。
Hein and Prüss [J. Differential Equations, 261(2016)4709-4727] presented a version of Hartman-Grobman type $C^{0}$ linearization result for semilinear hyperbolic evolution equations. They showed that the linearising map (homomorphism) and its inverse are Hölder continuous. An important question: is it possible to improve the regularity of the homomorphisms? In the present paper, we prove that if the mild solutions of semilinear system are bounded, then the regularity of the homomorphisms is Lipchitzian, but the inverse is merely Hölder continuous. We also give a generalized local linearization result in this paper. Finally, some applications end the paper. As pointed out by Backes [J. Differential Equations, 297 (2021) 536-574], even if the diffeomorphism $F$ is $C^{\infty}$, the homomorphism can fail to be locally Lipschitz. The homomorphisms are in general only locally Hölder continuous. However, by establishing two effective dichotomy integral inequalities, we prove that the conjugacy is Lipchitzian, but the inverse is Hölder continuous. Our result is the first one to observe the higher regularity of homomorphisms in the Hartman-Grobman theorem.
