Blow-up results for viscoelastic wave equations with weak damping. (June 2018)

• Record Type: Journal Article
• Authors: Park, Sun-Hye; Lee, Mi Jin; Kang, Jum-Ran
• Is Part Of: Applied mathematics letters. Volume 80(2018)
• Page Start: 20
• Page End: 26
• Publication Date: 2018-06
• Subjects: Blow-up -- Wave equation -- Viscoelasticity; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2018.01.002
• Languages: English
• ISSNs: 0893-9659

Abstract

Abstract: In this work we consider a viscoelastic wave equation of the form u t t − Δ u + ∫ 0 t g ( t − s ) Δ u ( s ) d s + h ( u t ) = | u | p − 2 u with Dirichlet boundary condition. There are much literature on the blow-up result of solutions for the wave equation with damping term having polynomial growth near zero. However, to my knowledge, there is no blow-up result of solutions for the viscoelastic wave equation without polynomial growth near zero assumption on the damping term. This work is devoted to study a finite time blow-up result of solution with nonpositive initial energy as well as positive initial energy without imposing any restrictive growth near zero assumption on the damping term.