Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations

Zhijian Yang; Yanan Li

doi:10.3934/dcdsb.2019036

Article Contents

2019, Volume 24, Issue 9: 4899-4912. Doi: 10.3934/dcdsb.2019036

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Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations

Zhijian Yang^, and
Yanan Li

School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou, 450001, China

^* Corresponding author: Zhijian Yang
^* Corresponding author: Zhijian Yang

Received: February 28, 2018

Revised: September 30, 2018

Early access: February 2019

Published: July 2019

This work is supported by National Natural Science Foundation of China (No.11671367)

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Abstract

The paper investigates the upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with structural damping: $ u_{tt}-M(\|\nabla u\|^2)\Delta u+(-\Delta)^\alpha u_t+f(u) = g(x,t) $, where $ \alpha\in(1/2, 1) $ is said to be a dissipative index. It shows that when the nonlinearity $ f(u) $ is of supercritical growth $ p: 1 \leq p< p_{\alpha}\equiv\frac{N+4\alpha}{(N-4\alpha)^+} $, the related evolution process has a pullback attractor for each $ \alpha\in(1/2, 1) $, and the family of pullback attractors is upper semicontinuous with respect to $ \alpha $. These results extend those in [27] for autonomous Kirchhoff wave models.

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Mathematics Subject Classification: Primary: 37L15, 37B55, 35B41; Secondary: 35B33, 35B65, 35L05.

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