Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator

Lianwang Deng

doi:10.3934/cpaa.2020009

Article Contents

2020, Volume 19, Issue 1: 145-174. Doi: 10.3934/cpaa.2020009

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Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator

Lianwang Deng

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

Received: October 2018

Revised: January 2019

Published: July 2019

This work was supported by the National Natural Science Foundations of China No. 11431008 and No. 11871041

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Abstract

We show the existence and $ C^{k, \gamma} $ smoothness of local integral manifolds at an equilibrium point for nonautonomous and ill-posed equations with sectorially dichotomous operator, provided that the nonlinearities are $ C^{k, \gamma} $ smooth with respect to the state variable. $ C^{k, \gamma} $ local unstable integral manifold follows from $ C^{k, \gamma} $ local stable integral manifold by reversing time variable directly. As an application, an elliptic PDE in infinite cylindrical domain is discussed.

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Mathematics Subject Classification: Primary: 37D10, 47J06, 34D09; Secondary: 47D06, 37L05.

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