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Blow‐up of solutions for a viscoelastic wave equation with variable exponents
Author(s) -
Park SunHye,
Kang JumRan
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5501
Subject(s) - mathematics , viscoelasticity , variable (mathematics) , wave equation , mathematical analysis , nonlinear system , energy (signal processing) , constant (computer programming) , function (biology) , physics , thermodynamics , statistics , quantum mechanics , evolutionary biology , biology , computer science , programming language
In this paper, we consider a viscoelastic wave equation with variable exponents:u t t − Δ u + ∫ 0 t g ( t − s ) Δ u ( s ) d s + a | u t | m ( x ) − 2u t = b | u | p ( x ) − 2 u , where the exponents of nonlinearity p (·) and m (·) are given functions and a , b > 0 are constants. For nonincreasing positive function g , we prove the blow‐up result for the solutions with positive initial energy as well as nonpositive initial energy. We extend the previous blow‐up results to a viscoelastic wave equation with variable exponents.