Article Contents

2023, Volume 28, Issue 9: 4735-4760. Doi: 10.3934/dcdsb.2023038

This issue Previous Article Approximate properties of stochastic functional differential equations with singular perturbations Next Article Equivalence of stability among stochastic differential equations, stochastic differential delay equations, and their corresponding Euler-Maruyama methods

Global existence, blow-up and optimal decay for a nonlinear viscoelastic equation with nonlinear damping and source term

Zaiyun Zhang^1,2, , and
Qiancheng Ouyang^1,

1.
School of Mathematics, Hunan Institute of Science and Technology, Yueyang, 414006, Hunan, China
2.
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China

^*Corresponding author: zhangzaiyun1226@126.com
^*Corresponding author: zhangzaiyun1226@126.com

Received: September 2022

Revised: January 2023

Early access: February 2023

Published: September 2023

Abstract / Introduction Full Text(HTML) Figure(5) Related Papers Cited by

Abstract

In this paper, we are concerned with a viscoelastic wave equation with memory term, nonlinear damping and source term. Firstly, using the potential well method combined with Galerkin approximation procedure, the global weak solutions are obtained. Secondly, we investigate the blow-up of solutions with initial positive and negative energy, as well as our result improves the earlier ones in [29] and [36]. Finally, under some assumptions imposed on damping coefficient and the relaxation function, we establish the optimal decay of the solutions which conducted by perturbed energy method. Moreover, we obtain that the exponential form of relaxation function lead to better decay result and memory term can slow down the energy decay by displaying the energy decay graphically.

Keywords:

Mathematics Subject Classification: 35A01, 35B40.

Citation:

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Figure 1. Graph of $ A(y) $

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Figure 2. Energy decay of Example 5.6

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Figure 3. Energy decay of Example 5.7

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Figure 4. Exponential decay of (93) compared with Fig. 2

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Figure 5. Polynomial decay of (94) compared with Fig. 3

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