Blow-up of solutions for a nonlinear viscoelastic wave equation with initial data at arbitrary energy level. Issue 12 (10th September 2019)

• Record Type: Journal Article
• Authors: Sun, Fenglong; Liu, Lishan; Wu, Yonghong
• Is Part Of: Applicable analysis. Volume 98:Issue 12(2019)
• Page Start: 2308
• Page End: 2327
• Publication Date: 2019-09-10
• Subjects: Wave equation -- viscoelasticity -- nonlinear damping -- supercritical source -- blow-up; 35L10 -- 35L20 -- 35L70 -- 35B44; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2018.1460812
• Languages: English
• ISSNs: 0003-6811

Abstract

ABSTRACT: In this paper, we study a three-dimensional (3D) viscoelastic wave equation with nonlinear weak damping, supercritical sources and prescribed past history u 0 ( x, t ), t ≤ 0 : u tt - k ( 0 ) Δ u - ∫ 0 + ∞ k ′ ( s ) Δ u ( t - s ) d s + | u t | m - 1 u t = | u | p - 1 u, in Ω × ( 0, T ), where the relaxation function k is monotone decreasing with k ( + ∞ ) = 1, m ≥ 1 and 1 ≤ p < 6 . When the source is stronger than dissipations, i.e. p > max { m, k ( 0 ) }, we obtain some finite time blow-up results with positive initial energy. In particular, we obtain the existence of certain solutions which blow up in finite time for initial data at arbitrary energy level.