Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (6): 1684-1704.

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Global Existence and General Decay for a Nonlinear Viscoelastic Equation with Time-Varying Delay and Velocity-Dependent Material Density

Zaiyun Zhang1,*(),Zhenhai Liu2,Youjun Deng3   

  1. 1 School of Mathematics, Hunan Institute of Science and Technology, Hunan Yueyang 414006
    2 College of Science, Guangxi University for Nationalities, Nanning 530006
    3 School of Mathematics and Statistics, Central South University, Changsha 410083
  • Received:2020-08-06 Online:2021-12-26 Published:2021-12-02
  • Contact: Zaiyun Zhang
  • Supported by:
    the NSFC(12071413);the NSFC(11971487);the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie(823731CONMECH);the Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering of Changsha University of Science and Technology Grant(2018MMAEZD05);the Scientific Research Fund of Hunan Provincial Education Department(18A325);the NSF of Guangxi(2018GXNSFDA138002);the NSF of Hunan Province(2021JJ30297);the NSF of Hunan Province(2020JJ2038);the Research and Innovation Team of Hunan Institute of Science and Technology(2019-TD-15);the Open Project of Hainan Key Laboratory of Computing Science and Application(JSKX201905)


In this paper, we investigate a nonlinear viscoelastic equation with a time-varying delay effect and velocity-dependent material density. Under suitable assumptions on the relaxation function and time-varying delay effect, we prove the global existence of weak solutions and general decay of the energy by using Faedo-Galerkin method and the perturbed energy method respectively. This result improves earlier ones in the literature, such as Refs.[1, 48-50].

Key words: Global existence, Nonlinear viscoelastic equation, General decay, Time-varying delay, Velocity-dependent material density, Perturbed energy method

CLC Number: 

  • O175.2