Dynamics for the damped wave equations on time-dependent domains

Feng Zhou; Chunyou Sun; Xin Li

doi:10.3934/dcdsb.2018068

Article Contents

2018, Volume 23, Issue 4: 1645-1674. Doi: 10.3934/dcdsb.2018068

This issue Previous Article Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions Next Article Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion

Dynamics for the damped wave equations on time-dependent domains

a.
College of Science, China University of Petroleum (East China), Qingdao, 266580, China
b.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China
c.
College of Science, Yanshan University, Qinhuangdao, 066004, China

* Corresponding author: Feng Zhou
* Corresponding author: Feng Zhou

Received: May 31, 2017

Revised: July 31, 2017

Early access: January 2018

Published: April 2018

The first author is supported by NSFC (Grants No. 11601522) and the Fundamental Research Funds for the Central Universities of China (No. 17CX02036A), the second author is supported by NSFC (Grants Nos. 11471148,11522109).

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Abstract

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains $\mathcal{O}_{t}$ in $\mathbb{R}^{3}$ are obtained from a bounded base domain $\mathcal{O}$ by a $C^{3}$-diffeomorphism $r(·, t)$. Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

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Mathematics Subject Classification: Primary: 35L70, 35L90; Secondary: 37L30.

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