Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$

Pengyan Ding; Zhijian Yang

doi:10.3934/cpaa.2019040

Article Contents

2019, Volume 18, Issue 2: 825-843. Doi: 10.3934/cpaa.2019040

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Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$

Pengyan Ding^1, and
Zhijian Yang^2, ,

1.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
2.
School of Mathematics and Statistics, Zhengzhou University, No. 100, Science Road, Zhengzhou 450001, China

* Corresponding author
* Corresponding author

Received: February 28, 2018

Revised: May 31, 2018

Published: October 2018

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Abstract

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on $\mathbb{R}^N$: $u_{tt}-φ(x)Δ u_{t}-φ(x)M(\|\nabla u\|^{2})Δ u+f(u) = h(x)$. It proves that when the growth exponent $p$ of the nonlinearity $f(u) $ is up to the supercritical range: $ 1≤ p < p^{**}(\equiv \frac{N+4}{(N-4)^+})$, the related solution semigroup has in weighted energy space a (strong) global attractor and a partially strong exponential attractor, respectively. In particular, the partially strong exponential attractor becomes the strong one in non-supercritical case (i.e., $1≤ p≤ p^{*}(\equiv \frac{N+2}{N-2})$).

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Mathematics Subject Classification: Primary: 37L30, 35B33, 35B40; Secondary: 35B41, 35L30.

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