Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels

Kun-Peng Jin; Jin Liang; Ti-Jun Xiao

doi:10.3934/dcdss.2021077

Article Contents

2021, Volume 14, Issue 9: 3141-3166. Doi: 10.3934/dcdss.2021077

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Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels

1.
School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
3.
Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

^* Corresponding author: Ti-Jun Xiao
^* Corresponding author: Ti-Jun Xiao

Received: August 31, 2020

Revised: December 31, 2020

Early access: June 2021

Published: September 2021

The second author is supported by NSF of China Grant No. 11971306. The third author is supported by NSF of China Grant No. 11771091 and No. 11831011

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Abstract

We are concerned with the polynomial stability and the integrability of the energy for second order integro-differential equations in Hilbert spaces with positive definite kernels, where the memory can be oscillating or sign-varying or not locally absolutely continuous (without any control conditions on the derivative of the kernel). For this stability problem, tools from the theory of existing positive definite kernels can not be applied. In order to solve the problem, we introduce and study a new mathematical concept – generalized positive definite kernel (GPDK). With the help of GPDK and its properties, we obtain an efficient criterion of the polynomial stability for evolution equations with such a general but more complicated and useful memory. Moreover, in contrast to existing positive definite kernels, GPDK allows us to directly express the decay rate of the related kernel.

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Mathematics Subject Classification: Primary: 35L05, 45N05; Secondary: 45M10, 35B35, 35B40.

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