Global existence and asymptotic behaviour of solution for a damped nonlinear hyperbolic equation. (September 2020)

• Record Type: Journal Article
• Authors: Pang, Tiantian; Shen, Jihong
• Is Part Of: Nonlinear analysis. Volume 198(2020)
• Publication Date: 2020-09
• Subjects: Fourth order nonlinear hyperbolic equation -- Global existence -- Asymptotic behaviour; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111885
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the initial–boundary value problem for a damped nonlinear hyperbolic equation u t t + Δ 2 u + α Δ 2 u t + Δ f ( Δ u ) = 0, x ∈ Ω, t > 0, u ( x, 0 ) = u 0 ( x ), u t ( x, 0 ) = u 1 ( x ), x ∈ Ω, u = 0 Δ u = 0, x ∈ ∂ Ω, t ≥ 0, where Ω ⊂ R n, n ≥ 1 and α is a positive constant. Under some assumptions on f ( s ) and initial data, we prove the global existence of solution. Furthermore, we prove that the solution decays to zero exponentially.