Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $

Pengyan Ding; Zhijian Yang

doi:10.3934/cpaa.2021006

Article Contents

2021, Volume 20, Issue 3: 1059-1076. Doi: 10.3934/cpaa.2021006

This issue Previous Article Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation Next Article Rational limit cycles of Abel equations

Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $

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

1.
College of Science, Henan University of Technology, Zhengzhou, 450001, China
2.
School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou, 450001, China

^* Corresponding author
^* Corresponding author

Received: April 30, 2020

Revised: October 31, 2020

Early access: January 2021

Published: March 2021

The research is supported by National Natural Science Foundation of China (Grant No. 11671367) and the Doctor Foundation of Henan University of Technology, China (No. 2019BS041). The first author is supported by the Doctor Foundation of Henan University of Technology, China (Grant No. 2019BS041). The second author is supported by NSFC (Grant No. 11671367)

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Abstract

The paper investigates the well-posedness and the existence of global attractor for a strongly damped wave equation on $ \mathbb{R}^{N} (N\geqslant 3): u_{tt}-\Delta u_{t}-\Delta u+u_{t}+u+g(u) = f(x) $. It shows that when the nonlinearity $ g(u) $ is of supercritical growth $ p $, with $ \frac{N+2}{N-2}\equiv p^*< p< p^{**} \equiv\frac{N+4}{(N-4)^+} $, (i) the initial value problem of the equation is well-posed and its weak solution possesses additionally partial regularity as $ t>0 $; (ii) the related solution semigroup has a global attractor in natural energy space. By using a new double truncation method on frequency space $ \mathbb{R}^N $ rather than approximating physical space $ \mathbb{R}^N $ by a sequence of balls $ \Omega_R $ as usual, we break through the longstanding existed restriction on this topic for $ p: 1\leqslant p\leqslant p^* $.

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Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 35B33, 35B65.

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