Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions

Nguyen Thanh Long; Hoang Hai Ha; Le Thi Phuong Ngoc; Nguyen Anh Triet

doi:10.3934/cpaa.2020023

Article Contents

2020, Volume 19, Issue 1: 455-492. Doi: 10.3934/cpaa.2020023

This issue Previous Article On the Schrödinger-Debye system in compact Riemannian manifolds Next Article The weak maximum principle for second-order elliptic and parabolic conormal derivative problems

Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions

1.
Department of Mathematics and Computer Science, VNUHCM-University of Science, 227 Nguyen Van Cu Str., Dist.5, Ho Chi Minh City, Vietnam
2.
Faculty of Applied Science, Ho Chi Minh City University of Technology, Vietnam National University Ho Chi Minh City, 268 Ly Thuong Kiet Str., Dist. 10, Ho Chi Minh City, Vietnam
3.
University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam
4.
Department of Mathematics, University of Architecture of Ho Chi Minh City, 196 Pasteur Str., Dist. 3, Ho Chi Minh City, Vietnam

^* Corresponding author
^* Corresponding author

Received: December 31, 2018

Revised: April 30, 2019

Published: July 2019

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Abstract

This paper is devoted to the study of a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Based on the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we first establish two local existence theorems of weak solutions. By the construction of a suitable Lyapunov functional, we next prove a blow up result and a decay result of global solutions.

Keywords:

System of nonlinear viscoelastic equations,
nonlinear boundary conditions,
Faedo-Galerkin method,
blow-up and exponential decay estimates.

Mathematics Subject Classification: Primary: 35L05, 35L15, 35L20, 35L55, 35L70.

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References

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